Twist formulas for one-row colored $A_2$ webs and $\mathfrak{sl}_3$ tails of $(2,2m)$-torus links
Wataru Yuasa
math.GT math.QA 57K14, 57K16, 11F27





The 𝔰𝔩_3 colored Jones polynomial J_λ^𝔰𝔩_3(L) is obtained by coloring the link components with two-row Young diagram λ. 
Although it is difficult to compute J_λ^𝔰𝔩_3(L) in general, we can calculate it by using Kuperberg's A_2 skein relation.
In this paper, 
we show some formulas for twisted two strands colored by one-row Young diagram in A_2 web space and compute J_(n,0)^𝔰𝔩_3(T(2,2m)) for an oriented (2,2m)-torus link. 
These explicit formulas derives the 𝔰𝔩_3 tail of T(2,2m). 
They also gives explicit descriptions of the 𝔰𝔩_3 false theta series with one-row coloring because the 𝔰𝔩_2 tail of T(2,2m) is known as the false theta series.

[
    [
    Received: date / Accepted: date
===================================











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    triple/.style args=[#1] in [#2] in [#3]
        #1,preaction=preaction=draw,#3,draw,#2
    








§ INTRODUCTION

In this paper, 
we will make tools to calculate the quantum invariant of knots and links obtained from 𝔰𝔩_3 by using the Kuperberg's linear skein theory. 
Especially, we will prove a full twist formula which is useful to compute explicitly the 𝔰𝔩_3 colored Jones polynomial.
The quantum invariants of knots are obtained through a functor from the category of framed oriented tangles to the representation category of a quantum group of a simple Lie algebra.
Thus, one can define an invariant J_V^𝔤(K) of a knot K for each simple Lie algebra 𝔤 and its representation V, that is, there exist infinitely many quantum invariants of knots.
However, it is difficult to compute J_V^𝔤(K) explicitly. 
The most simple case is 𝔤=𝔰𝔩_2 with the 2-dimensional irreducible representation V=ℂ^2.
This quantum invariant reconstructs the Jones polynomial J(K).
We know that J(K) can be calculated from a knot diagram of K by using the Kauffman bracket skein relation, for example <cit.>. 
More generally, one can compute the colored Jones polynomial J_n(K) obtained from the (n+1)-dimensional irreducible representation of 𝔰𝔩_2 by applying the Kauffman bracket skein relation to a knot diagram colored by the Jones-Wenzl projector.
The Jones-Wenzl projector <cit.> corresponds to the projector onto the (n+1)-dimensional irreducible representation in V^⊗ n.
In this case, J_n(K) is explicitly calculated for many knots and there are many useful formulas which decompose a tangle diagram to the linear sum of the web basis by the skein relation.
This method to calculate quantum invariants from knot diagrams is called the linear skein theory, see for example <cit.>.
The linear skein theory is constructed for some other 𝔤 than 𝔰𝔩_2.
In a calculational point of view,
one of the most reasonable quantum invariants next after 𝔰𝔩_2 is the 𝔰𝔩_3 colored Jones polynomial J_(m,n)^𝔰𝔩_3(K) which is obtained from the irreducible representation with the highest weight (m,n) of 𝔰𝔩_3.
We use the linear skein theory for 𝔰𝔩_3 constructed by Kuperberg <cit.>,
see Definition <ref> of the A_2 skein relation.
However, there are few non-trivial examples of explicit formulas of J_(m,n)^𝔰𝔩_3(K).
For example, Lawrence <cit.> calculated J_(m,n)^𝔰𝔩_3(K) for the trefoil knot, more generally, In <cit.> for the (2,2m+1)- and (4,5)-torus knots by using a representation theoretical method.
In <cit.>, the author calculated J_(n,0)^𝔰𝔩_3(K) for the two-bridge links K with one-row coloring (n,0) by the Kuperberg's linear skein theory. 
Important formulas used in <cit.> is the following full twist formulas:


[anti-parallel full twist formula <cit.>]
Let d=min{s,t} and δ=|s-t|.

    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,5);
    [-<-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,5) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,5) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
        =q^st/3∑_l=0^∞
    q^l^2-lq^-(s+t)l
    (q)_ls  l_qt  l_q [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [-<-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [-<-=.5] (-6,4) to[out=east, in=east] (-6,-4);
    [->-=.5] (6,4) to[out=west, in=west] (6,-4);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,6)[above]t-l;
     at (0,-6)[below]s-l;
     at (-6,0)l;
     at (6,0)l;
     
       =q^-2/3d(d+δ)-d∑_k=0^d
    q^k(k+δ)+k(q)_d+δ/(q)_k+δd  k_q [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [-<-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [-<-=.5] (-6,4) to[out=east, in=east] (-6,-4);
    [->-=.5] (6,4) to[out=west, in=west] (6,-4);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,6)[above]k+t-d;
     at (0,-6)[below]k+s-d;
     at (-4,0)[left]d-k;
     at (4,0)[right]d-k;
     .


From this theorem, the author obtained an m-full twist formula for anti-parallel two strands.
[anti-parallel m-full twist formula <cit.>]
Let k=(k_1,…,k_m) be an m-tuple of integers, k_0=d=min{s,t}, and δ=| s-t |.

    [baseline=-.6ex][xshift=-1cm]
     
    (-.5,.4) – +(-.2,0)
    (-.5,-.4) – +(-.2,0);
    [->-=1, white, double=black, double distance=0.4pt, ultra thick] 
    (-.5,-.4) to[out=east, in=west] (.0,.4);
    [-<-=1, white, double=black, double distance=0.4pt, ultra thick] 
    (-.5,.4) to[out=east, in=west] (.0,-.4);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (0,-.4) to[out=east, in=west] (.5,.4);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (0,.4) to[out=east, in=west] (.5,-.4);
    [fill=white] (-.5,-.6) rectangle +(-.1,.4);
    [fill=white] (-.5,.6) rectangle +(-.1,-.4);
     at (-.5,-.6)[left]s;
     at (-.5,.6)[left]t;
     at (.0,.0)⋯;
     at (.0,-.4)[below]m full twists;
    [xshift=1cm]
    
    (.5,-.4) – +(.2,0)
    (.5,.4) – +(.2,0);
    [->-=1, white, double=black, double distance=0.4pt, ultra thick] 
    (-.5,-.4) to[out=east, in=west] (.0,.4);
    [-<-=1, white, double=black, double distance=0.4pt, ultra thick] 
    (-.5,.4) to[out=east, in=west] (.0,-.4);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (0,-.4) to[out=east, in=west] (.5,.4);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (0,.4) to[out=east, in=west] (.5,-.4);
    [fill=white] (.5,-.6) rectangle +(.1,.4);
    [fill=white] (.5,.6) rectangle +(.1,-.4);
     at (.5,-.6)[right]s;
     at (.5,.6)[right]t;
        =q^-2m/3k_0(k_0+δ)-2mk_0∑_k_0≥ k_1≥⋯≥ k_m≥ 0 C(k)
     [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [-<-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [-<-=.5] (-6,4) to[out=east, in=east] (-6,-4);
    [->-=.5] (6,4) to[out=west, in=west] (6,-4);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,6)[above]k_m+(t-k_0);
     at (0,-6)[below]k_m+(s-k_0);
     at (-4,0)[left]k_0-k_m;
     at (4,0)[right]k_0-k_m;
     ,

where

    C(k)=q^∑_i=1^mk_i(k_i+δ)+2k_iq^k_0-k_m(q)_k_0+δ/(q)_k_m+δk_0 k_1',k_2',…,k_m',k_m_q

 and k_i+1'=k_i-k_i+1 for i=0,1,…,m-1.


The above formula plays an important role in the study of 𝔰𝔩_3 tails of knots and links.
The tail of a knot K is a q-series which is a limit of the colored Jones polynomials {J_n(K;q)}_n. 
Independently, the existence of the tails for alternating knots was shown in <cit.> and <cit.>, more generally, for adequate links in <cit.>. 
In <cit.>, Garoufalidis and Lê showed a more general stability, the existence of the tail is the zero-stability, for alternating knots.
Some explicit descriptions of tails are known for T(2,2m+1) in <cit.>, 
for T(2,2m) in <cit.>, 
for a pretzel knot P(2k+1,2,2l+1) in <cit.>, 
for knots with small crossing numbers in <cit.>, <cit.>, and <cit.>.
Especially, the tail of T(2,2m+1) is given by the theta series and one of T(2,2m) is the false theta series.

One can consider a tail for the 𝔰𝔩_3 colored Jones polynomial. 
We call it the 𝔰𝔩_3 tail.
However, there are many problems such that how do we define a limit for {J_(k,l)^𝔰𝔩_3(K)}_k,l and the existence of the 𝔰𝔩_3 tail etc.
As a case study, the author gave an explicit formula of the 𝔰𝔩_3 tail for the (2,2m)-torus link T_⇆(2,2m) with anti-parallel orientation by using Theorem <ref> in <cit.>, see also Theomrem <ref>.
This 𝔰𝔩_3 tail can be considered a special type of the 𝔰𝔩_3 false theta series.
In fact, Bringmann-Kaszian-Milas <cit.> commented that the 𝔰𝔩_3 tail of T_⇆(2,2m) coincides with the diagonal part of the 𝔰𝔩_3 false theta series defined through the study of vertex operator algebras in <cit.>.

This paper is organized as follows.
In section 2, we review the Kuperberg's linear skein theory fo A_2 and give a diagrammatic definition of the Jones-Wenzl projector for A_2.
In section 3, we derive formulas which decompose an m-full twist of two-strands with one-row coloring into the linear sum of A_2 basis webs.
We use the generating function for Young diagrams by involving the decompositions of the A_2 web with lattice paths in proofs of these formulas.
As an application, we compute the 𝔰𝔩_3 colored Jones polynomials and the 𝔰𝔩_3 tails of oriented (2,m)-torus links with one-row coloring in sectoin 4. 





§ DEFINITION AND PROPERTIES OF THE A_2 CLASP

We use the following notation.


  * [n]=q^n/2-q^-n/2/q^1/2-q^1/2 is a quantum integer for n∈ℤ_≥ 0.

  * n  k=[n]/[k][n-k] for 0≤ k≤ n and n  k=0 for k>n.

  * (q)_n=∏_i=1^n(1-q^i) is a q-Pochhammer symbol.

  * nk_q=(q)_n/(q)_k(q)_n-k for 0≤ k≤ n and nk_q=0 for k>n.

  * nk_1,k_2,…,k_m_q=(q)_n/(q)_k_1(q)_k_2⋯(q)_k_m for positive integers k_i's such that ∑_i=1^mk_i=n.


Let us define A_2 web spaces based on <cit.>. 
We consider a disk D with signed marked points (P,ϵ) on its boundary where P⊂∂ D is a finite set and ϵ P→{+,-} a map. 

A tangled bipartite uni-trivalent graph on D is an immersion of a directed graph into D satisfying (1) – (4):


  * the valency of a vertex of underlying graph is 1 or 3,

  * all crossing points are transversal double points of two edges with under/over information,

  * the set of univalent vertices coincides with P,

  * a neighborhood of a vertex is one of the followings:


   
 [baseline=-.6ex, scale=.1]
[dashed, fill=white] (0,0) circle [radius=7];
[-<-=.5] (0:0) – (90:7); 
[-<-=.5] (0:0) – (210:7); 
[-<-=.5] (0:0) – (-30:7);

 
,
 [baseline=-.6ex, scale=.1]
[dashed, fill=white] (0,0) circle [radius=7];
[->-=.5] (0:0) – (90:7); 
[->-=.5] (0:0) – (210:7); 
[->-=.5] (0:0) – (-30:7);

 
,
 [baseline=-.6ex, scale=.1]
[dashed] (0,0) circle [radius=7];
[-<-=.5] (0,0) – (7,0);
at (0,0) [left]+;
[thick] (0,7) – (0,-7);
[fill=cyan] (0,0) circle [radius=.5];
 
,
 [baseline=-.6ex, scale=.1]
[dashed] (0,0) circle [radius=7];
[->-=.5] (0,0)–(7,0);
at (0,0) [left]-;
[thick] (0,7) – (0,-7);
[fill=cyan] (0,0) circle [radius=.5];
 
.



[A_2 web space <cit.>]
Let G(ϵ;D) be the set of boundary fixing isotopy classes of tangled trivalent graphs on D.
The A_2 web space W(ϵ;D) is the quotient of the ℂ[q^1/6,q^-1/6]-vector space on G(ϵ;D) by the following A_2 skein relation:


  * [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[->-=.8] (-45:7) – (135:7);
[->-=.8, white, double=black, double distance=0.4pt, line width=2.4pt] (-135:7) – (45:7);
 
=q^1/3 [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[->-=.5] (-45:7) to[out=north west, in=south] (3,0) to[out=north, in=south west] (45:7);
[->-=.5] (-135:7) to[out=north east, in=south] (-3,0) to[out=north, in=south east] (135:7);
 
-q^-1/6 [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[->-=.5] (-45:7) – (0,-3);
[->-=.5] (-135:7) – (0,-3);
[-<-=.5] (45:7) – (0,3);
[-<-=.5] (135:7) – (0,3);
[-<-=.5] (0,-3) – (0,3);,

  * [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[->-=.8] (-135:7) – (45:7);
[->-=.8, white, double=black, double distance=0.4pt, line width=2.4pt] (-45:7) – (135:7);
 
=q^-1/3 [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[->-=.5] (-45:7) to[out=north west, in=south] (3,0) to[out=north, in=south west] (45:7);
[->-=.5] (-135:7) to[out=north east, in=south] (-3,0) to[out=north, in=south east] (135:7);
 
-q^1/6 [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[->-=.5] (-45:7) – (0,-3);
[->-=.5] (-135:7) – (0,-3);
[-<-=.5] (45:7) – (0,3);
[-<-=.5] (135:7) – (0,3);
[-<-=.5] (0,-3) – (0,3);

  * [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[-<-=.6] (-45:7) – (-45:3);
[->-=.6] (-135:7) – (-135:3);
[->-=.6] (45:7) – (45:3);
[-<-=.6] (135:7) – (135:3);
[-<-=.5] (45:3) – (135:3);
[->-=.5] (-45:3) – (-135:3);
[-<-=.5] (45:3) – (-45:3);
[->-=.5] (135:3) – (-135:3);
 
=
 [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[-<-=.5] (-45:7) to[out=north west, in=south] (3,0) to[out=north, in=south west] (45:7);
[->-=.5] (-135:7) to[out=north east, in=south] (-3,0) to[out=north, in=south east] (135:7);
 
+
 [baseline=-.6ex, rotate=90, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[-<-=.5] (-45:7) to[out=north west, in=south] (3,0) to[out=north, in=south west] (45:7);
[->-=.5] (-135:7) to[out=north east, in=south] (-3,0) to[out=north, in=south east] (135:7);,


  * [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[->-=.5] (0,-7) – (0,-3);
[->-=.5] (0,3) – (0,7);
[-<-=.5] (0,-3) to[out=east, in=south] (3,0) to[out=north, in=east] (0,3);
[-<-=.5] (0,-3) to[out=west, in=south] (-3,0) to[out=north, in=west] (0,3);
 
=
[2]
 [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[->-=.5] (0,-7) – (0,7);,


  * [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[->-=.5] (0,0) circle [radius=3];
 
=
[3]
 [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
 
=
 [baseline=-.6ex, scale=.1][thin, dashed, fill=white] (0,0) circle [radius=7];
[-<-=.5] (0,0) circle [radius=3];
 
.

An element in W(ϵ;D) is called web and an element in G(ϵ;D) without crossings which has no internal 0-, 2-, 4-gons a basis web. 
Any web is described as the sum of basis webs.


The A_2 skein relation realize the Reidemeister moves (R1) – (R4), that is,
we can show that webs represent diagrams in the left side and right side is the same web in W(D;ϵ).

    (R1)  [baseline=-.6ex, scale=.1] [thin, dashed] (0,0) circle [radius=5];
     (3,-2)
    to[out=south, in=east] (2,-3)
    to[out=west, in=south] (0,0)
    to[out=north, in=west] (2,3)
    to[out=east, in=north] (3,2);
    [white, double=black, double distance=0.4pt, line width=2.4pt] (0,-5) 
    to[out=north, in=west] (2,-1)
    to[out=east, in=north] (3,-2);
    [white, double=black, double distance=0.4pt, line width=2.4pt] (3,2)
    to[out=south, in=east] (2,1)
    to[out=west, in=south] (0,5);
     [baseline=-.6ex, scale=.5] [<->, xshift=1.5cm] (1,0)–(2,0);
     [baseline=-.6ex, scale=.1, xshift=3cm][thin, dashed] (0,0) circle [radius=5];
     (90:5) to (-90:5);
           (R2)  [baseline=-.6ex, scale=.1][thin, dashed] (0,0) circle [radius=5];
     (135:5) to[out=south east, in=west] (0,-2) to[out=east, in=south west](45:5);
    [white, double=black, double distance=0.4pt, line width=2.4pt]
    (-135:5) to[out=north east, in=west] (0,2) to[out=east, in=north west] (-45:5);
     [baseline=-.6ex, scale=.5] [<->, xshift=1.5cm] (1,0)–(2,0);
     [baseline=-.6ex, scale=.1, xshift=3cm][thin, dashed] (0,0) circle [radius=5];
     (135:5) to[out=south east, in=west](0,2) to[out=east, in=south west](45:5);
     (-135:5) to[out=north east, in=west](0,-2) to[out=east, in=north west] (-45:5);
     ,
       (R3)  [baseline=-.6ex, scale=.1] [thin, dashed] (0,0) circle [radius=5];
     (-135:5) – (45:5);
    [white, double=black, double distance=0.4pt, line width=2.4pt] (135:5) – (-45:5);
    [white, double=black, double distance=0.4pt, line width=2.4pt] 
    (180:5) to[out=east, in=west](0,3) to[out=east, in=west] (-0:5);
     [baseline=-.6ex, scale=.5][<->, xshift=1.5cm] (1,0)–(2,0);
     [baseline=-.6ex, scale=.1, xshift=3cm] [thin, dashed] (0,0) circle [radius=5];
     (-135:5) – (45:5);
     [white, double=black, double distance=0.4pt, line width=2.4pt] (135:5) – (-45:5);
    [white, double=black, double distance=0.4pt, line width=2.4pt] 
    (180:5) to[out=east, in=west] (0,-3) to[out=east, in=west] (-0:5);
     ,
          (R4)  [baseline=-.6ex, scale=.1] [thin, dashed] (0,0) circle [radius=5];
     (0:0) – (90:5); 
     (0:0) – (210:5); 
     (0:0) – (-30:5);
    [white, double=black, double distance=0.4pt, line width=2.4pt]
    (180:5) to[out=east, in=west] (0,3) to[out=east, in=west] (0:5);
     [baseline=-.6ex, scale=.5][<->, xshift=1.5cm] (1,0)–(2,0);
     [baseline=-.6ex, scale=.1] [thin, dashed] (0,0) circle [radius=5];
     (0:0) – (90:5); 
     (0:0) – (210:5); 
     (0:0) – (-30:5);
    [white, double=black, double distance=0.4pt, line width=2.4pt]
    (180:5) to[out=east, in=west] (0,-3) to[out=east, in=west](0:5);
     , 
     [baseline=-.6ex, scale=.1] [thin, dashed] (0,0) circle [radius=5];
    [white, double=black, double distance=0.4pt, line width=2.4pt]
    (180:5) to[out=east, in=west] (0,3) to[out=east, in=west] (0:5);
    [white, double=black, double distance=0.4pt, line width=2.4pt] (0:0) – (90:5); 
     (0:0) – (210:5); 
     (0:0) – (-30:5);
     [baseline=-.6ex, scale=.5][<->, xshift=1.5cm] (1,0)–(2,0);
     [baseline=-.6ex, scale=.1] [thin, dashed] (0,0) circle [radius=5];
    [white, double=black, double distance=0.4pt, ultra thick]
    (180:5) to[out=east, in=west] (0,-3) to[out=east, in=west](0:5);
     (0:0) – (90:5); 
    [white, double=black, double distance=0.4pt, line width=2.4pt] (0:0) – (210:5); 
    [white, double=black, double distance=0.4pt, line width=2.4pt] (0:0) – (-30:5);
     .




We review a diagrammatic definition of an A_2 clasp introduced in <cit.> and its properties. 
The A_2 clasp gives a coloring of strands in a web by pairs of non-negative integers. 
It plays an important role as is the case with the Jones-Wenzl projector.



We construct a projector called the A_2 clasp of type (n,m) in a special web space (+^n-^m,+^n-^m). 

[the Temperley-Lieb category for A_2]
Let D=[0,1]×[0,1] and 𝐧 denotes a set of n points on I=[0,1] dividing it into n+1 equal parts.
The Temperley-Lieb category  is a linear category over ℂ(q^1/6) is defined as follows:


  * an object is a word (finite sequence) over {+,-},

  * the tensor product of two words is defined by the product (concatenation),

  * the space of morphisms (α,β) is the web space W(α̅⊔β;D) where α̅ is the word consisting of opposite signs of α. 

We identify an object α with length n as a map 𝐧→{+,-} by using the order on 𝐧. 
A map α̅⊔β means that the domain 𝐧⊂ I of the map α̅ is identified with the top edge [0,1]×{0} and β identified with the bottom edge [0,1]×{1}. 


  * The composition GF∈ G(α̅⊔γ;D) of F∈ G(α̅⊔β;D) and G∈ G(β̅⊔γ;D) is given by gluing the top edge of F and the bottom edge of G.

  * The “tensor product” F⊗ G∈ G(α̅_1α̅_2,β_1β_2;D) of F∈ G(α̅_1⊔β_1;D) and G∈ G(α_2⊔β_2;D) by gluing the right edge {1}×[0,1] of F and the left edge {0}×[0,1] of G.

They define the composition and the tensor product on the space of morphisms by linearization. 
The diagrammatic description of them is in Figure <ref>.




 is generated by identity morphisms id_+∈(+,+), id_-∈(-,-) and the following morphisms.

    t_-^++
    =
     [baseline=-.6ex, scale=.1][thick] (-8,-5) rectangle (8,5);
    [->-=.5] (0,0) – (-5,5);
    [->-=.5] (0,0) – (5,5);
    [->-=.5] (0,0) – (0,-5);
     , 
    t_+^–
    =
     [baseline=-.6ex, scale=.1][thick] (-8,-5) rectangle (8,5);
    [-<-=.5] (0,0) – (-5,5);
    [-<-=.5] (0,0) – (5,5);
    [-<-=.5] (0,0) – (0,-5);
     , 
    t_–^+
    =
     [baseline=-.6ex, scale=.1, yscale=-1][thick] (-8,-5) rectangle (8,5);
    [->-=.5] (0,0) – (-5,5);
    [->-=.5] (0,0) – (5,5);
    [->-=.5] (0,0) – (0,-5);
     , 
    t_++^-
    =
     [baseline=-.6ex, scale=.1, yscale=-1][thick] (-8,-5) rectangle (8,5);
    [-<-=.5] (0,0) – (-5,5);
    [-<-=.5] (0,0) – (5,5);
    [-<-=.5] (0,0) – (0,-5);
     ,
       b^-+
    =
     [baseline=-.6ex, scale=.1][thick] (-8,-5) rectangle (8,5);
    [->-=.5] (-5,5) to[out=south, in=west] (0,1) to[out=east, in=south] (5,5);
     , 
    b^+-
    =
     [baseline=-.6ex, scale=.1][thick] (-8,-5) rectangle (8,5);
    [-<-=.5] (-5,5) to[out=south, in=west] (0,1) to[out=east, in=south] (5,5);
     , 
    d_+-
    =
     [baseline=-.6ex, scale=.1, yscale=-1][thick] (-8,-5) rectangle (8,5);
    [->-=.5] (-5,5) to[out=south, in=west] (0,1) to[out=east, in=south] (5,5);
     , 
    d_-+
    =
     [baseline=-.6ex, scale=.1, yscale=-1][thick] (-8,-5) rectangle (8,5);
    [-<-=.5] (-5,5) to[out=south, in=west] (0,1) to[out=east, in=south] (5,5);
     .




Let us define one colored A_2 clasp m0 before defining mn. 
We do not describe the boundary of D in what follows.

[The A_2 clasp in (+^m,+^m)]
The A_2 clasp m0 described by a white box with m∈ℤ_≥ 0 is defined as follows.


  * [baseline=-.6ex, scale=.1][->-=.8] (0,-6) – (0,6);
[fill=white] (-3,-1) rectangle (3,1);
 at (0,5) [left]0;
 
=∅ (the empty diagram),
[baseline=-.6ex, scale=.1][->-=.8] (0,-6) – (0,6);
[fill=white] (-3,-1) rectangle (3,1);
 at (0,5) [left]1;
 
=
 [baseline=-.6ex, scale=.1][->-=.5] (0,-6) – (0,6); 
 
=id_+∈(+,+)


  * [baseline=-.6ex, scale=.1][->-=.8] (0,-6) – (0,6);
[fill=white] (-3,-1) rectangle (3,1);
 at (0,5) [left]m+1;
 
= 
 [baseline=-.6ex, scale=.1][->-=.8] (0,-6) – (0,6);
[->-=.5] (5,-6) – (5,6);
[fill=white] (-2,-1) rectangle (2,1);
 at (0,5) [left]m;
 
-[n]/[n+1] [baseline=-.6ex, scale=.1][->-=.6] (0,4) – (0,7);
[->-=.6] (0,-7) – (0,-4);
[->-=.5] (-2,-3) – (-2,3);
[-<-=.7] (5,-1) – (5,1);
[-<-=.7] (2,3) to[out=south, in=west] (5,1);
[-<-=.3] (8,7) – (8,3) to[out=south, in=east] (5,1);
[->-=.7, yscale=-1] (2,3) to[out=south, in=west] (5,1);
[->-=.3, yscale=-1] (8,7) – (8,3) to[out=south, in=east] (5,1);
[fill=white] (-3,3) rectangle (3,4);
[fill=white] (-3,-3) rectangle (3,-4);
 at (0,6) [left]m;
 at (0,-6) [left]m;
 at (-2,0) [left]m-1;
 ∈(+^n+1,+^n+1)

The A_2 clasp in 0m∈(-^m,-^m) is also defined in the same way.

We introduce the A_2 clasp mn in (+^m-^n,+^m-^n) based on <cit.>.
[The A_2 clasp in (+^m-^n,+^m-^n)]

    mn
    =
     [baseline=-.6ex, scale=.1][->-=.2, ->-=.8] (-2,-5) – (-2,5);
    [-<-=.2, -<-=.8] (2,-5) – (2,5);
    [fill=white] (-4,-1) rectangle (4,1);
     at (-2,5) [left]m;
     at (-2,-5) [left]m;
     at (2,5) [right]n;
     at (2,-5) [right]n;
     
    =
    ∑_i=0^min{m,n}
    (-1)^i
    m in i/m+n+1 i [baseline=-.6ex, scale=.1][->-=.5] (-4,5) – (-4,8);
    [-<-=.5] (4,5) – (4,8);
    [-<-=.5, yscale=-1] (-4,5) – (-4,8);
    [->-=.5, yscale=-1] (4,5) – (4,8);
    [->-=.5] (-5,-4) – (-5,4);
    [-<-=.5] (5,-4) – (5,4);
    [-<-=.5] (-3,4) to[out=south, in=south] (3,4);
    [->-=.5, yscale=-1] (-3,4) to[out=south, in=south] (3,4);
    [fill=white] (-6,4) rectangle (-2,5);
    [fill=white, xscale=-1] (-6,4) rectangle (-2,5);
    [fill=white, yscale=-1] (-6,4) rectangle (-2,5);
    [fill=white, xscale=-1, yscale=-1] (-6,4) rectangle (-2,5);
     at (-5,0) [left]m-i;
     at (5,0) [right]n-i;
     at (0,5) i;
     at (0,-5) i;
     at (-4,7) [left]m;
     at (4,7) [right]n;
     at (-4,-7) [left]m;
     at (4,-7) [right]n;
     .



The A_2 clasp has the following well-known properties

Let m, n are non-negative integers.


  * (kl)(mn)=(mn)(kl)=mn for 0≤ k≤ m and 0≤ l≤ n.

  * F(mn)=0 if F∈{t_–^+, t_++^-, d_+-, d_-+}.

  * (mn)F=0 if F∈{t_-^++, t_+^–, b^-+, b^+-}.

In the above, we omit tensor components of identity morphisms in F and kl.


Next, we construct a special web JW_α^β∈(α,β)  which has a similar properties to the A_2 clasp where α and β are obtained by rearranging the order of +^m-^n.

Let us introduce two basis webs called H-webs:

    H_+-^-+
    =
     [baseline=-.6ex, scale=.1][->-=.3,-<-=.8] (0,-4) – (0,4);
    [-<-=.3,->-=.8] (5,-4) – (5,4);
    [-<-=.5] (0,0) – (5,0);
     
    ,
       H_-+^+-
    =
     [baseline=-.6ex, scale=.1][-<-=.3,->-=.8] (0,-4) – (0,4);
    [->-=.3,-<-=.8] (5,-4) – (5,4);
    [->-=.5] (0,0) – (5,0);
     .


Let α∈ be a object obtained by rearranging the order of +^m-^n.
Then, one can construct a morphism from +^m-^n to α by composing H-webs.
σ(α) denotes such a web minimizing the number of H-webs and it is uniquely determined.
In the same way, 
one can construct the morphism from α to +^m-^n by pre-composing H-webs and denote it by τ(α).
We remark that σ(α) (resp. τ(α)) does not contain H_-+^+- (resp.H_+-^-+).

Let α and β be words obtained by rearranging +^m-^n. 
We define the following webs:

    mn^α   =σ(α)(mn)∈(+^m-^n,α), 
    JW_β^+^m-^n   =(mn)τ(β)∈(β,+^m-^-), 
    JW_β^α   =mn^αJW_β^+^m-^n∈(β,α).

Let us describe JW_β^α diagrammatically by id_β∘(the white box)∘id_α.


F(mn^α)=0 if F∈{t_–^+, t_++^-, d_+-, d_-+} and (JW_β^+^m-^n)F=0 if F∈{t_-^++, t_+^–, b^-+, b^+-}.


We only show the first case and denote σ(α) by σ for simplicity
We give a proof by induction on the number h(σ) of H_+-^-+ contained in σ.
If h(σ)=0, it is clear since mn^α=mn. 
If h(σ)=1, then σ=id_+^m-1⊗ H_+-^-+⊗id_-^n-1 and one can prove easily.
When h(σ)=k+1, 
we choose α'∈ such that σ(α)=(id⊗ H_+-^-+⊗id)σ(α').
Then,
σ is one of the following webs:

    type I:   [baseline=-.6ex, scale=.1] (10,0) – (10,6);
     (-10,0) – (-10,6);
    [-<-=.5] (6,0) – (6,6);
    [->-=.5] (-6,0) – (-6,6);
    [->-=.3,-<-=.8] (-4,0) – (-4,6);
    [-<-=.3,->-=.8] (4,0) – (4,6);
    [-<-=.5] (-4,3) – (4,3);
    [fill=lightgray] (-12,0) rectangle (12,-4);
     at (8,5) ⋯;
     at (-8,5) ⋯;
     at (0,-2) σ';
     ,
           type II:   [baseline=-.6ex, scale=.1] (10,0) – (10,6);
     (-10,0) – (-10,6);
    [->-=.5] (6,0) – (6,6);
    [->-=.5] (-6,0) – (-6,6);
    [->-=.3,-<-=.8] (-4,0) – (-4,6);
    [-<-=.3,->-=.8] (4,0) – (4,6);
    [-<-=.5] (-4,3) – (4,3);
    [fill=lightgray] (-12,0) rectangle (12,-4);
     at (8,5) ⋯;
     at (-8,5) ⋯;
     at (0,-2) σ';
     ,
        type III:   [baseline=-.6ex, scale=.1] (10,0) – (10,6);
     (-10,0) – (-10,6);
    [->-=.5] (6,0) – (6,6);
    [-<-=.5] (-6,0) – (-6,6);
    [->-=.3,-<-=.8] (-4,0) – (-4,6);
    [-<-=.3,->-=.8] (4,0) – (4,6);
    [-<-=.5] (-4,3) – (4,3);
    [fill=lightgray] (-12,0) rectangle (12,-4);
     at (8,5) ⋯;
     at (-8,5) ⋯;
     at (0,-2) σ';
     ,
           type IV:   [baseline=-.6ex, scale=.1] (10,0) – (10,6);
     (-10,0) – (-10,6);
    [-<-=.5] (6,0) – (6,6);
    [-<-=.5] (-6,0) – (-6,6);
    [->-=.3,-<-=.8] (-4,0) – (-4,6);
    [-<-=.3,->-=.8] (4,0) – (4,6);
    [-<-=.5] (-4,3) – (4,3);
    [fill=lightgray] (-12,0) rectangle (12,-4);
     at (8,5) ⋯;
     at (-8,5) ⋯;
     at (0,-2) σ';
     ,

where σ'=σ(α').
Because of h(σ')=k, 
mn^α' satisfies the statement in Lemma <ref> by the induction hypothesis.
We only need to calculate a web obtained by multiplication of F at the part we pictured in the above.

In the case of type I, the possibility of F is d_+- or d_-+. 
Then, the vanishing of F(σmn) is clear.

In the case of type II, 
it is similar to type I if F=d_+- or d_-+.
We only have to consider F=t_++^-, that is, show

    [baseline=-.6ex, scale=.1] (4,6) – (6,6);
    [-<-=.7] (5,6) – (5,9);
     (10,0) – (10,6);
     (-10,0) – (-10,6);
    [->-=.5] (6,0) – (6,6);
    [->-=.5] (-6,0) – (-6,6);
    [->-=.3,-<-=.8] (-4,0) – (-4,6);
    [-<-=.3,->-=.8] (4,0) – (4,6);
    [-<-=.5] (-4,3) – (4,3);
    [fill=lightgray] (-12,0) rectangle (12,-4);
     at (8,5) ⋯;
     at (-8,5) ⋯;
     at (0,-2) σ';
     ∘mn=0.

The construction of σ and α'=⋯++-+⋯ say that there exists σ” such that

    [baseline=-.6ex, scale=.1] (4,6) – (6,6);
    [-<-=.7] (5,6) – (5,9);
    
     (10,0) – (10,6);
     (-10,0) – (-10,6);
    [->-=.5] (6,0) – (6,6);
    [->-=.5] (-6,0) – (-6,6);
    [->-=.3,-<-=.8] (-4,0) – (-4,6);
    [-<-=.3,->-=.8] (4,0) – (4,6);
    [-<-=.5] (-4,3) – (4,3);
    [fill=lightgray] (-12,-2) rectangle (12,-6);
    
     (10,0) – (10,-2);
     (-10,0) – (-10,-2);
     (6,0) – (6,-2);
     (-6,0) – (-6,-2);
     (-4,0) – (-4,-2);
     (4,0) – (4,-2);
    [-<-=.8] (4,0) – (6,0);
     at (8,5) ⋯;
     at (-8,5) ⋯;
     at (0,-4) σ”;
     
    =
     [baseline=-.6ex, scale=.1]
     (10,0) – (10,6);
     (-10,0) – (-10,6);
    [-<-=.5] (6,0) – (6,6);
    [->-=.5] (-6,0) – (-6,6);
    [->-=.3,-<-=.8] (-4,0) – (-4,6);
     (4,0) – (4,3);
    [-<-=.5] (-4,3) – (4,3);
    [fill=lightgray] (-12,-2) rectangle (12,-6);
    
     (10,0) – (10,-2);
     (-10,0) – (-10,-2);
     (6,0) – (6,-2);
     (-6,0) – (-6,-2);
     (-4,0) – (-4,-2);
     (4,0) – (4,-2);
     at (8,5) ⋯;
     at (-8,5) ⋯;
     at (0,-4) σ”;
     
    +
     [baseline=-.6ex, scale=.1]
     (10,0) – (10,6);
     (-10,0) – (-10,6);
    [->-=.5] (-6,0) – (-6,6);
    [->-=.3,-<-=.8] (-4,0) – (-4,6);
     (4,3) – (4,6);
    [-<-=.5] (-4,3) – (4,3);
    [fill=lightgray] (-12,-2) rectangle (12,-6);
    
     (10,0) – (10,-2);
     (-10,0) – (-10,-2);
     (6,0) – (6,-2);
     (-6,0) – (-6,-2);
     (-4,0) – (-4,-2);
     (4,0) – (4,-2);
    [-<-=.8] (4,0) – (6,0);
     at (8,5) ⋯;
     at (-8,5) ⋯;
     at (0,-4) σ”;

The first term of the RHS has t_++^- and the second d_-+ on the top of σ”. 
Because of h(σ”)=k-1 and the induction hypothesis, 
the composition of the RHS with mn vanish.
One can prove type III and type IV in the same way.


Let α, β, and γ in  be obtained by rearranging +^m-^n.
Then, JW_β^γJW_α^β=JW_α^γ. 


It is easy to see by Lemma <ref> because τ(β) is the reflection of σ(β) at the horizontal line with the oppositely directed edges.



Let us introduce two types of a web, “stair-step” and “triangle”.
These webs also appear in <cit.>.

For positive integers n and m, 
[baseline=-.6ex, scale=.1] (-4,0) – (-2,0);
 (2,0) – (4,0);
 (0,-4) – (0,-2);
 (0,2) – (0,4);
[fill=white] (-2,-2) rectangle (2,2);
 (-2,2) – (2,-2);
 at (-4,0) [left]n;
 at (4,0) [right]n;
 at (0,4) [above]m;
 at (0,-4) [below]m;
is defined by

    [baseline=-.6ex, scale=.1] (-4,0) – (-2,0);
     (2,0) – (4,0);
     (0,-4) – (0,-2);
     (0,2) – (0,4);
    [fill=white] (-2,-2) rectangle (2,2);
     (-2,2) – (2,-2);
     at (-4,0) [left]n;
     at (4,0) [right]n;
     at (0,4) [above]1;
     at (0,-4) [below]1;
     
    =
     [baseline=-.6ex, scale=.1] (-7,-5) – (7,-5);
     (-7,-3) – (7,-3);
     (-7,5) – (7,5);
     (4,5) – (4,7);
     (3,3) – (3,5);
     (-4,-7) – (-4,-5);
     (-3,-5) – (-3,-3);
     (-2,-3) – (-2,-1);
    [yscale=2.5] at (-9,0) {;
     at (-9,0) [left]n;
    [rotate=90] at (-6,1)⋯;
    [rotate=90] at (6,1)⋯;
    [rotate=45] at (0,1)⋯;
       and
        [baseline=-.6ex, scale=.1] (-4,0) – (-2,0);
     (2,0) – (4,0);
     (0,-4) – (0,-2);
     (0,2) – (0,4);
    [fill=white] (-2,-2) rectangle (2,2);
     (-2,2) – (2,-2);
     at (-4,0) [left]n;
     at (4,0) [right]n;
     at (0,4) [above]m;
     at (0,-4) [below]m;
     
    =
     [baseline=-.6ex, scale=.1] (-4,0) – (-2,0);
     (2,0) – (4,0);
     (0,-4) – (0,-2);
     (0,2) – (0,4);
    [fill=white] (-2,-2) rectangle (2,2);
     (-2,2) – (2,-2);
     at (-4,0) [left]n;
     at (0,4) [above]m-1;
     at (0,-4) [below]m-1;
    [xshift=8cm]
     (-4,0) – (-2,0);
     (2,0) – (4,0);
     (0,-4) – (0,-2);
     (0,2) – (0,4);
    [fill=white] (-2,-2) rectangle (2,2);
     (-2,2) – (2,-2);
     at (-4,0) [above]n;
     at (4,0) [right]n;
     at (0,4) [above]1;
     at (0,-4) [below]1;
       for   m>1.

Specifying a direction on an edge around the box determine the all directions of edges in the box.




For positive integer n,
[baseline=-.6ex, scale=.1] (30:5) – (0,0);
 (150:5) – (0,0);
 (270:5) – (0,0);
[fill=white] (-30:4) – (90:4) – (210:4) – cycle;
 at (30:5) [below]n;
 at (150:5) [below]n;
 at (270:5) [left]n;
is defined by
[baseline=-.6ex, scale=.1] (30:5) – (0,0);
 (150:5) – (0,0);
 (270:5) – (0,0);
[fill=white] (-30:4) – (90:4) – (210:4) – cycle;
 at (30:5) [below]1;
 at (150:5) [below]1;
 at (270:5) [left]1;
 
= [baseline=-.6ex, scale=.1] (30:5) – (0,0);
 (150:5) – (0,0);
 (270:5) – (0,0); and 
[baseline=-.6ex, scale=.1] (30:5) – (0,0);
 (150:5) – (0,0);
 (270:5) – (0,0);
[fill=white] (-30:4) – (90:4) – (210:4) – cycle;
 at (30:5) [below]n;
 at (150:5) [below]n;
 at (270:5) [left]n;
 
= [baseline=-.6ex, scale=.1] (-10,5) – (5,5);
 (-5,-4) – (-5,5);
 (-10,1) – (5,1);
 (0,-4) – (0,1);
[fill=white] (-30:3) – (90:3) – (210:3) – cycle;
[fill=white] (-6,-2) rectangle (-4,3);
 (-6,3) – (-4,-2);
 at (-10,5) [left]1;
 at (5,5) [right]1;
 at (-5,-4) [below]1;
 at (-10,0) [below]n-1;
 at (4,0) [above]n-1;
 at (0,-4) [below]n-1;
for n>1.
We obtain a web by specifying a direction of an edge around the triangle.







We note a graphical description of properties of A_2 clasp and useful formulas.


    [baseline=-.6ex, scale=.1] (0,-6) – (0,6);
    [fill=white] (-4,-4) rectangle (4,-2);
    [fill=white] (-4,4) rectangle (4,2);
     at (0,5) [left]γ;
     at (0,0) [left]β;
     at (0,-5) [left]α;
     
    =
     [baseline=-.6ex, scale=.1] (0,-6) – (0,6);
    [fill=white] (-4,-1) rectangle (4,1);
     at (0,4) [left]γ;
     at (0,-4) [left]α;
     
    ,
       [baseline=-.6ex, scale=.1] (-2,1) – (0,3);
     (2,1) – (0,3);
     (0,6) – (0,3);
     (0,-6) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (4,3) …;
     at (-4,3) …;
     =0
    ,
       [baseline=-.6ex, scale=.1] (-2,1) to[out=north, in=west] (0,3) to[out=east, in=north] (2,1);
     (0,-6) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (4,3) …;
     at (-4,3) …;
     =0
    ,
        [baseline=-.6ex, scale=.1][-<-=.5] (-3,1) to[out=north, in=south west] (0,4);
    [->-=.5] (3,1) to[out=north, in=south east] (0,4);
    [-<-=.8] (0,4) to[out=north east, in=south] (3,7);
    [->-=.8] (0,4) to[out=north west, in=south] (-3,7);
     (0,-5) – (0,0);
    [fill=white] (0,2) – (-2,4) – (0,6) – (2,4) – cycle;
     (0,2) – (0,6);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-3,7) [above]m;
     at (3,7) [above]n;
     
    =
     [baseline=-.6ex, scale=.1][->-=.5] (-2,1) – (-2,7);
    [-<-=.5] (2,1) – (2,7);
     (0,-5) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-2,7) [above]m;
     at (2,7) [above]n;
     
    ,
       [baseline=-.6ex, scale=.1][->-=.5] (-3,1) to[out=north, in=south west] (0,4);
    [-<-=.5] (3,1) to[out=north, in=south east] (0,4);
    [->-=.8] (0,4) to[out=north east, in=south] (3,7);
    [-<-=.8] (0,4) to[out=north west, in=south] (-3,7);
     (0,-5) – (0,0);
    [fill=white] (0,2) – (-2,4) – (0,6) – (2,4) – cycle;
     (0,2) – (0,6);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-3,7) [above]m;
     at (3,7) [above]n;
     
    =
     [baseline=-.6ex, scale=.1][-<-=.5] (-2,1) – (-2,7);
    [->-=.5] (2,1) – (2,7);
     (0,-5) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-2,7) [above]m;
     at (2,7) [above]n;
     
    ,
        [baseline=-.6ex, scale=.1][-<-=.8, white, double=black, double distance=0.4pt, ultra thick] (-3,1) to[out=north, in=south] (3,7);
    [->-=.8, white, double=black, double distance=0.4pt, ultra thick] (3,1) to[out=north, in=south] (-3,7);
     (0,-5) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-3,7) [above]m;
     at (3,7) [above]n;
     
    =(-q^-1/6)^mn [baseline=-.6ex, scale=.1][->-=.5] (-2,1) – (-2,7);
    [-<-=.5] (2,1) – (2,7);
     (0,-5) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-2,7) [above]m;
     at (2,7) [above]n;
     
    ,
       [baseline=-.6ex, scale=.1][->-=.8, white, double=black, double distance=0.4pt, ultra thick] (3,1) to[out=north, in=south] (-3,7);
    [-<-=.8, white, double=black, double distance=0.4pt, ultra thick] (-3,1) to[out=north, in=south] (3,7);
     (0,-5) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-3,7) [above]m;
     at (3,7) [above]n;
     
    =(-q^1/6)^mn [baseline=-.6ex, scale=.1][->-=.5] (-2,1) – (-2,7);
    [-<-=.5] (2,1) – (2,7);
     (0,-5) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-2,7) [above]m;
     at (2,7) [above]n;
     
    ,
        [baseline=-.6ex, scale=.1][->-=.8, white, double=black, double distance=0.4pt, ultra thick] (-3,1) to[out=north, in=south] (3,7);
    [->-=.8, white, double=black, double distance=0.4pt, ultra thick] (3,1) to[out=north, in=south] (-3,7);
     (0,-5) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-3,7) [above]m;
     at (3,7) [above]n;
     
    =q^mn/3 [baseline=-.6ex, scale=.1][->-=.5] (-2,1) – (-2,7);
    [->-=.5] (2,1) – (2,7);
     (0,-5) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-2,7) [above]m;
     at (2,7) [above]n;
     
    ,
       [baseline=-.6ex, scale=.1][->-=.8, white, double=black, double distance=0.4pt, ultra thick] (3,1) to[out=north, in=south] (-3,7);
    [->-=.8, white, double=black, double distance=0.4pt, ultra thick] (-3,1) to[out=north, in=south] (3,7);
     (0,-5) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-3,7) [above]m;
     at (3,7) [above]n;
     
    =q^-mn/3 [baseline=-.6ex, scale=.1][->-=.5] (-2,1) – (-2,7);
    [->-=.5] (2,1) – (2,7);
     (0,-5) – (0,0);
    [fill=white] (-6,-1) rectangle (6,1);
     at (5,4) …;
     at (-5,4) …;
     at (-2,7) [above]m;
     at (2,7) [above]n;
     ,
        [baseline=-.6ex, scale=.1] (-8,0) – (8,0);
     (0,-7) – (0,0);
    [fill=white] (-4,-3) – (0,4) – (4,-3) – cycle;
    [fill=white] (-3,-5) rectangle (3,-4);
    [fill=white] (-6,-3) rectangle (-5,3);
    [fill=white] (6,-3) rectangle (5,3);
     at (-8,0) [above]n;
     at (8,0) [above]n;
     at (0,-7) [right]n;
     
    = [baseline=-.6ex, scale=.1] (-8,0) – (6,0);
     (0,-7) – (0,0);
    [fill=white] (-4,-3) – (0,4) – (4,-3) – cycle;
    [fill=white] (-3,-5) rectangle (3,-4);
    [fill=white] (-6,-3) rectangle (-5,3);
     at (-8,0) [above]n;
     at (6,0) [above]n;
     at (0,-7) [right]n;



Graphical interpretation of the previous definitions and lemmas. 
One can prove formulas related to the braiding and the triangle by easy calculation, see <cit.>.




§ TWIST FORMULA

In this section, we show new twist formulas derived from a combinatorial method related to the generating function of Young diagrams. 
We firstly explain this method which is used in the proof of full twist formulas for the Kauffman bracket and Theorem <ref> in <cit.>.

Let {σ_n(k,l)| 0≤ k,l≤ n} be a set of webs satisfying the following conditions:


  * σ_n(k,l) is basis web if k+l=n,

  * σ_n(k,l)=X(k,l)σ_n(k+1,l)+Y(k,l)σ_n(k,l+1),

  * X(k,l)Y(k+1,l)=qY(k,l)X(k,l+1),

where X(k,l), Y(k,l)∈ℂ(q^1/6).



    σ_n(0,0)=∑_k+l=n∏_j=0^l-1Y(0,j)∏_i=0^k-1X(i,l)nk_qσ_n(k,l).



Let us label a lattice point (k,l) in ℤ×ℤ with σ(k,l), and a directed edge from (k,l) to (k+1,l) with X(k,l), and a directed edge from (k,l) to (k,l+1) with Y(k,l).
We denote the label of an edge e by w(e).
In the expansion of σ_n(0,0), it is seen that the coefficient of σ_n(k,l), for k+l=n, is 

    ∑_γ(∏_e∈γw(e))=∑_a_i∈{X,Y} a_1a_2⋯ a_n,

where X (resp. Y) appears k (resp. l) times in a_1a_2⋯ a_n and γ is a path from (0,0) to (k,l) which consists of the above directed edges.
Although we describe X and Y without its coordinate in the above, it is uniquely determined. 
For example, a_1a_2a_3a_4⋯=XYYX⋯=X(0,0)Y(1,0)Y(1,1)X(1,2)⋯. 
By the condition (3), the RHS of (<ref>) is 

    Y^lX^k∑ q^#{i<j| a_i=X, a_j=Y}=Y^lX^knk_q.


In <cit.>, we used Young diagrams instead of sequences of X and Y. 
However, it is intrinsically the same because a lattice path from (0,0) to (k,l) is interpreted as a Young diagram.
The formula of Theorem <ref> is obtained from Proposition <ref> by setting n=d=min{s,t} and 

    σ_n(k,l)=
    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,4);
    [-<-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,6) to[out=east, in=west] (0,-4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-4) to[out=east, in=west] (6,6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,4) to[out=east, in=west] (6,-5);
    [->-=.5] (-6,-4) to[out=east, in=south] (-5,0) to[out=north, in=east] (-6,4);
    [-<-=.5] (6,-4) to[out=west, in=south] (5,0) to[out=north, in=west] (6,4);
    [->-=.5] (-6,-6) – (6,-6);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,-6)[below]k;
     at (5,0)[right]l;
     at (-5,0)[left]l;
     .



In a similar way, We will prove the following formula.
[parallel full twist formula]
Let d=min{s,t} and δ=| s-t |.

    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,5) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,5) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]t;
     at (-7,7)[left]s;
     at (7,-7)[right]t;
     at (7,7)[right]s;
        =q^2/3st∑_l=0^∞
    q^l^2-l/2q^-(s+t)l
    (q)_ls  l_qt  l_q [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (-3,0) – (0,3) – (3,0) – (0,-3) – cycle;
     (0,3) – (0,-3);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,6)[above]t-l;
     at (0,-6)[below]s-l;
     at (-5,4)[below]l;
     at (-5,-4)[above]l;
     at (5,4)[below]l;
     at (5,-4)[above]l;
     
       =q^-d(d+δ)/3-d/2∑_k=0^d
    q^k(k+δ)+k/2(q)_d+δ/(q)_k+δd  k_q [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (-3,0) – (0,3) – (3,0) – (0,-3) – cycle;
     (0,3) – (0,-3);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,6)[above]k+t-d;
     at (0,-6)[below]k+s-d;
     at (-3,2)[left]d-k;
     at (-3,-2)[left]d-k;
     at (3,2)[right]d-k;
     at (3,-2)[right]d-k;
     .



To prove this formula by Proposition <ref>,
we set 

    σ_d(k,l)=
    [baseline=-.6ex, scale=.1] (-17,5) – (-14,5);
     (-17,-5) – (-14,-5);
     (17,5) – (14,5);
     (17,-5) – (14,-5);
    [->-=.5] (-14,6) – (-6,6);
    [->-=.5] (-14,-6) – (-6,-6);
    [->-=.5] (-6,0) – (-11,0);
    [->-=.5] (-14,4) to[out=east, in=north east] (-11,0);
    [->-=.5] (-14,-4) to[out=east, in=south east] (-11,0);
    [fill=white] (-12,0) – (-9,2) – (-9,-2) – cycle;
    [-<-=.5] (14,6) – (6,6);
    [-<-=.5] (14,-6) – (6,-6);
    [-<-=.5] (6,0) – (11,0);
    [-<-=.5] (14,4) to[out=west, in=north west] (11,0);
    [-<-=.5] (14,-4) to[out=west, in=south west] (11,0);
    [fill=white] (12,0) – (9,2) – (9,-2) – cycle;
    
    [-<-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,-4) – (6,-4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-1) to[out=east, in=west] (0,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,6) to[out=east, in=west] (0,-1);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-1) to[out=east, in=west] (6,6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (6,-1);
    [fill=white] (-7,8) rectangle (-6,4);
    [fill=white] (-7,-8) rectangle (-6,2);
    [fill=white] (7,8) rectangle (6,4);
    [fill=white] (7,-8) rectangle (6,2);
     at (3,-5)[above]k;
     at (3,-5)[below]l;
     at (0,6)[above]s-k-l;
    
    [fill=white] (-14,-8) rectangle (-15,-2);
    [fill=white] (-14,8) rectangle (-15,2);
    [fill=white] (14,-8) rectangle (15,-2);
    [fill=white] (14,8) rectangle (15,2);
     at (-10,6)[above]t-l;
     at (-10,-6)[below]s-l;
     at (10,6)[above]t-l;
     at (10,-6)[below]s-l;
     at (-10,1)[above]l;
     at (-10,-1)[below]l;
     at (10,1)[above]l;
     at (10,-1)[below]l;
     at (-17,5)[above]t;
     at (-17,-5)[below]s;
     at (17,5)[above]t;
     at (17,-5)[below]s;
           if d=s,
       σ_d(k,l)=
    [baseline=-.6ex, scale=.1] (-17,5) – (-14,5);
     (-17,-5) – (-14,-5);
     (17,5) – (14,5);
     (17,-5) – (14,-5);
    [->-=.5] (-14,6) – (-6,6);
    [->-=.5] (-14,-6) – (-6,-6);
    [->-=.5] (-6,0) – (-11,0);
    [->-=.5] (-14,4) to[out=east, in=north east] (-11,0);
    [->-=.5] (-14,-4) to[out=east, in=south east] (-11,0);
    [fill=white] (-12,0) – (-9,2) – (-9,-2) – cycle;
    [-<-=.5] (14,6) – (6,6);
    [-<-=.5] (14,-6) – (6,-6);
    [-<-=.5] (6,0) – (11,0);
    [-<-=.5] (14,4) to[out=west, in=north west] (11,0);
    [-<-=.5] (14,-4) to[out=west, in=south west] (11,0);
    [fill=white] (12,0) – (9,2) – (9,-2) – cycle;
    
    [-<-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,4) – (6,4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-6) to[out=east, in=west] (0,1);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,1) to[out=east, in=west] (0,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-6) to[out=east, in=west] (6,1);
    [white, double=black, double distance=0.4pt, ultra thick] (0,1) to[out=east, in=west] (6,-6);
    [fill=white] (-7,-8) rectangle (-6,-4);
    [fill=white] (-7,8) rectangle (-6,-2);
    [fill=white] (7,-8) rectangle (6,-4);
    [fill=white] (7,8) rectangle (6,-2);
     at (3,5)[above]l;
     at (3,5)[below]k;
     at (0,-6)[below]t-k-l;
    
    [fill=white] (-14,-8) rectangle (-15,-2);
    [fill=white] (-14,8) rectangle (-15,2);
    [fill=white] (14,-8) rectangle (15,-2);
    [fill=white] (14,8) rectangle (15,2);
     at (-10,6)[above]t-l;
     at (-10,-6)[below]s-l;
     at (10,6)[above]t-l;
     at (10,-6)[below]s-l;
     at (-10,1)[above]l;
     at (-10,-1)[below]l;
     at (10,1)[above]l;
     at (10,-1)[below]l;
     at (-17,5)[above]t;
     at (-17,-5)[below]s;
     at (17,5)[above]t;
     at (17,-5)[below]s;
           if d=t.

We remark that it is easy to see that A_2 clasps in the middle of σ_d(k,l) can be removable from the definition of A_2 clasp and Lemma <ref>.



    σ_d(k,l)=q^2/3(d+δ-l)σ_d(k+1,l)+q^-1/6q^k+δ/3q^-2/3(d+δ-l)(1-q^d+δ-l)σ_d(k,l+1)


We can confirm that X(k,l)=q^2/3(d+δ-l) and Y(k,l)=q^-1/6q^k/3q^-2/3(n-l)(1-q^n-l) satisfy (2) of Proposition <ref>.
Thus,
we can prove Theorem <ref> by applying Proposition <ref>.
This equation is obtained as a sequel to a calculation of A_2 webs.
We will prove it in the case of d=s, but the otherwise is proven in the same way.


Let n and i are positive integers satisfying i<n.

    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,4) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,6) [above]i;
     at (0,-5) [below]n-i;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
     
    =q^2/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,4) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,6) [above]i+1;
     at (0,-5) [below]n-i-1;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
     
    +q^-1/2q^-n/3q^i/3(1-q)
     [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,4) – (-2,0);
    [->-=.5] (-6,-5) – (-2,0);
    [->-=.7] (2,0) – (6,4);
    [->-=.7] (2,0) – (6,-5);
    [-<-=.5] (-2,0) – (2,0);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]1;
     at (-7,7)[left]n;
     at (7,-7)[right]1;
     at (7,7)[right]n;
     at (0,6)[above]n-1;




We will make a red mark on a place where we use the skein relation and Lemma <ref>.

    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) to[out=east, in=west] (0,-2);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,2) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,2);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-2) to[out=east, in=west] (6,4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,4) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,1);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,1);
     at (0,6) [above]i;
     at (0,-2) [above]1;
     at (0,-5) [below]n-i-1;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
    [opacity=.2, magenta] (-3, 1) circle [radius=1];
    [opacity=.2, magenta] (3, 1) circle [radius=1];
     
       =q^2/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,4) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,6) [above]i+1;
     at (0,-5) [below]n-i-1;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
     
    +q^-5/6(1-q)
    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.4, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) – (-2,2);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,2) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,2);
    [->-=.8, white, double=black, double distance=0.4pt, ultra thick] (2,2) – (6,-5);
    [-<-=.5] (-2,2) – (2,2);
    [->-=.5] (-6,4) – (-2,2);
    [-<-=.5] (6,4) – (2,2);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,1);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,1);
     at (0,6) [above]i;
     at (0,-5) [below]n-i-1;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
    [opacity=.2, magenta] (-6, 1) rectangle (-5,5);
    [opacity=.2, magenta] (6, 1) rectangle (5,5);
     
       =q^2/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,4) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,6) [above]i+1;
     at (0,-5) [below]n-i-1;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
     
    +q^-5/6(1-q)q^2/3(n-i-1)[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-4) to[out=east, in=west] (4,4) – (6,4);
    [->-=.5] (-6,2) – (-4,0);
    [->-=.5] (-6,-5) – (-4,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-4,0) – (4,0);
    [-<-=.5] (6,2) – (4,0);
    [-<-=.5] (6,-5) – (4,0);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) – (-4,4) to[out=east, in=west] (0,-4);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,1);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,1);
     at (0,6) [above]i;
     at (0,-4) [below]n-i-1;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
    [opacity=.2, magenta] (-2, 0) circle [radius=1];
    [opacity=.2, magenta] (2, 0) circle [radius=1];
     
       =q^2/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,4) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,6) [above]i+1;
     at (0,-5) [below]n-i-1;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
     
    +q^-5/6(1-q)q^-2/3(n-i-1)(-q^1/6)^2(n-i-1)[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-4) to[out=east, in=west] (4,4) – (6,4);
    [->-=.5] (-6,2) – (-4,0);
    [->-=.5] (-6,-5) – (-4,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-4,0) – (4,0);
    [-<-=.5] (6,2) – (4,0);
    [-<-=.5] (6,-5) – (4,0);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) – (-4,4) to[out=east, in=west] (0,-4);
    [fill=white] (-1,-1) rectangle (-3,1);
    [fill=white] (1,-1) rectangle (3,1);
     (-1,-1) – (-3,1);
     (1,-1) – (3,1);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,1);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,1);
     at (0,6) [above]i;
     at (0,-4) [below]n-i-1;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
    [opacity=.2, magenta] (-3,-4) rectangle (3,2);
     
       =q^2/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,4) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,6) [above]i+1;
     at (0,-5) [below]n-i-1;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
     
    +q^-5/6(1-q)q^2/3(n-i-1)(-q^1/6)^2(n-i-1) [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,4) – (-2,0);
    [->-=.5] (-6,-5) – (-2,0);
    [->-=.7] (2,0) – (6,4);
    [->-=.7] (2,0) – (6,-5);
    [-<-=.5] (-2,0) – (2,0);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]1;
     at (-7,7)[left]n;
     at (7,-7)[right]1;
     at (7,7)[right]n;
     at (0,6)[above]n-1;







Let n be a positive integer.

    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,5) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,5) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
     
    =q^2n/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,5) – (6,5);
    [->-=.5] (-6,-5) – (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
     
    +q^-1/2q^-n/3(1-q^n)
     [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,4) – (-2,0);
    [->-=.5] (-6,-5) – (-2,0);
    [->-=.7] (2,0) – (6,4);
    [->-=.7] (2,0) – (6,-5);
    [-<-=.5] (-2,0) – (2,0);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]1;
     at (-7,7)[left]n;
     at (7,-7)[right]1;
     at (7,7)[right]n;
     at (0,6)[above]n-1;




By using Lemma <ref>, we obtain

    q^2i/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,4) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,6) [above]i;
     at (0,-5) [below]n-i;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
     
    +q^-1/2q^-n/3(1-q^i)
     [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,4) – (-2,0);
    [->-=.5] (-6,-5) – (-2,0);
    [->-=.7] (2,0) – (6,4);
    [->-=.7] (2,0) – (6,-5);
    [-<-=.5] (-2,0) – (2,0);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]1;
     at (-7,7)[left]n;
     at (7,-7)[right]1;
     at (7,7)[right]n;
     at (0,6)[above]n-1;
     
         =q^2(i+1)/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,4) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,4) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,6) [above]i+1;
     at (0,-5) [below]n-i-1;
     at (-7,-7) [left]1;
     at (-7,7) [left]n;
     at (7,-7) [right]1;
     at (7,7) [right]n;
     
    +q^-1/2q^-n/3(1-q^i+1)
     [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,4) – (-2,0);
    [->-=.5] (-6,-5) – (-2,0);
    [->-=.7] (2,0) – (6,4);
    [->-=.7] (2,0) – (6,-5);
    [-<-=.5] (-2,0) – (2,0);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]1;
     at (-7,7)[left]n;
     at (7,-7)[right]1;
     at (7,7)[right]n;
     at (0,6)[above]n-1;
     .

We repeatedly apply the above equation to the RHS of Lemma <ref> at i=0.



Let i and j be positive integers.

    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,5) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,5) to[out=east, in=west] (6,-5);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,5) [above]i;
     at (0,-5) [below]j;
     at (-7,-7) [left]i;
     at (-7,7) [left]j;
     at (7,-7) [right]i;
     at (7,7) [right]j;
     
    =q^2j/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-4) to[out=east, in=west] (0,5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,5) to[out=east, in=west] (0,-4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-4) to[out=east, in=west] (6,5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,5) to[out=east, in=west] (6,-4);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,5) [above]i-1;
     at (0,-4) [above]j;
     at (0,-6) [below]1;
     at (-7,-7) [left]i;
     at (-7,7) [left]j;
     at (7,-7) [right]i;
     at (7,7) [right]j;
     
    +q^-1/6q^-i+j/3(1-q^j)
    [baseline=-.6ex, scale=.1] (-17,5) – (-14,5);
     (-17,-5) – (-14,-5);
     (17,5) – (14,5);
     (17,-5) – (14,-5);
    [->-=.5] (-14,6) – (-6,6);
    [->-=.5] (-14,-6) – (-6,-6);
    [->-=.5] (-7,0) – (-11,0);
    [->-=.5] (-14,4) – (-11,0);
    [->-=.5] (-14,-4) – (-11,0);
    [-<-=.5] (14,6) – (6,6);
    [-<-=.5] (14,-6) – (6,-6);
    [-<-=.5] (7,0) – (11,0);
    [-<-=.5] (14,4) – (11,0);
    [-<-=.5] (14,-4) – (11,0);
    
    [-<-=.5] (-6,-6) – (6,-6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-1) to[out=east, in=west] (0,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,6) to[out=east, in=west] (0,-1);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-1) to[out=east, in=west] (6,6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (6,-1);
    [fill=white] (-7,8) rectangle (-6,4);
    [fill=white] (-7,-8) rectangle (-6,2);
    [fill=white] (7,8) rectangle (6,4);
    [fill=white] (7,-8) rectangle (6,2);
     at (3,-5)[below]1;
     at (0,6)[above]i-1;
    
    [fill=white] (-14,-8) rectangle (-15,-2);
    [fill=white] (-14,8) rectangle (-15,2);
    [fill=white] (14,-8) rectangle (15,-2);
    [fill=white] (14,8) rectangle (15,2);
     at (-10,6)[above]j-1;
     at (-10,-6)[below]i-1;
     at (10,6)[above]j-1;
     at (10,-6)[below]i-1;
     at (-17,5)[above]j;
     at (-17,-5)[below]i;
     at (17,5)[above]j;
     at (17,-5)[below]i;




By Lemma <ref>,

    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-6) to[out=east, in=west] (0,1);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-3) to[out=east, in=west] (0,5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,5) to[out=east, in=west] (0,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-6) to[out=east, in=west] (6,5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,5) to[out=east, in=west] (6,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (0,1) to[out=east, in=west] (6,-6);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (-7,-7) rectangle (-6,-2);
    [fill=white] (7,7) rectangle (6,3);
    [fill=white] (7,-7) rectangle (6,-2);
     at (0,-6) [above]j;
     at (0,1) [above]1;
     at (0,5) [above]i-1;
     at (-7,7) [left]j;
     at (-7,-7) [left]i;
     at (7,7) [right]j;
     at (7,-7) [right]i;
        =q^2j/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-4) to[out=east, in=west] (0,5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,5) to[out=east, in=west] (0,-4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-4) to[out=east, in=west] (6,5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,5) to[out=east, in=west] (6,-4);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,5) [above]i-1;
     at (0,-4) [above]j;
     at (0,-6) [below]1;
     at (-7,-7) [left]i;
     at (-7,7) [left]j;
     at (7,-7) [right]i;
     at (7,7) [right]j;
     
    +q^-1/2q^-j/3(1-q^j)
    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
     (0,2) to[out=east, in=west] (6,6);
     (0,0) – (3,0);
    [->-=.3, white, double=black, double distance=0.4pt, ultra thick] (3,0) – (6,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (-6,-3) – (-3,-3) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-6,6) to[out=east, in=west] (0,2);
    [-<-=.9, white, double=black, double distance=0.4pt, ultra thick] (-3,0) – (0,0);
    [->-=.2, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (3,-3) – (6,-3);
    [->-=.8, white, double=black, double distance=0.4pt, ultra thick] (-6,-6) – (-3,0);
    [-<-=.7] (6,4) – (3,0);
    [->-=.8] (-6,4) – (-3,0);
    [fill=white] (-7,-7) rectangle (-6,-2);
    [fill=white] (-7,7) rectangle (-6,2);
    [fill=white] (7,-7) rectangle (6,-2);
    [fill=white] (7,7) rectangle (6,2);
     at (2,6) [above]i-1;
     at (-4,6) [above]j-1;
     at (-7,-7) [left]i;
     at (-7,7) [left]j;
     at (7,-7) [right]i;
     at (7,7) [right]j;
    [opacity=.2, magenta] (-4,-3) circle[radius=1];
    [opacity=.2, magenta] (4,-3) circle[radius=1];
     
       =q^2j/3[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-4) to[out=east, in=west] (0,5);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,5) to[out=east, in=west] (0,-4);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-4) to[out=east, in=west] (6,5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,5) to[out=east, in=west] (6,-4);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (0,5) [above]i-1;
     at (0,-4) [above]j;
     at (0,-6) [below]1;
     at (-7,-7) [left]i;
     at (-7,7) [left]j;
     at (7,-7) [right]i;
     at (7,7) [right]j;
     
    +q^-1/2q^-j/3(1-q^j)q^-2/3(i-1)[baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
     (0,2) to[out=east, in=west] (6,6);
     (0,0) – (4,0);
    [->-=.7, white, double=black, double distance=0.4pt, ultra thick] (4,0) – (6,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-6,-6) – (-4,-6) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-6,6) to[out=east, in=west] (0,2);
    [-<-=.9, white, double=black, double distance=0.4pt, ultra thick] (-4,0) – (0,0);
    [->-=.2, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-6) – (6,-6);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-6,-3) – (-4,0);
    [-<-=.6] (6,3) – (4,0);
    [->-=.6] (-6,3) – (-4,0);
    [fill=white] (-7,-7) rectangle (-6,-2);
    [fill=white] (-7,7) rectangle (-6,2);
    [fill=white] (7,-7) rectangle (6,-2);
    [fill=white] (7,7) rectangle (6,2);
     at (2,6) [above]i-1;
     at (-4,6) [above]j-1;
     at (-7,-7) [left]i;
     at (-7,7) [left]j;
     at (7,-7) [right]i;
     at (7,7) [right]j;
     .

As we remarked before Proposition <ref>, 
the definition of A_2 clasp and its property says that

    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
     (0,2) to[out=east, in=west] (6,6);
     (0,0) – (4,0);
    [->-=.7, white, double=black, double distance=0.4pt, ultra thick] (4,0) – (6,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-6,-6) – (-4,-6) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-6,6) to[out=east, in=west] (0,2);
    [-<-=.9, white, double=black, double distance=0.4pt, ultra thick] (-4,0) – (0,0);
    [->-=.2, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-6) – (6,-6);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-6,-3) – (-4,0);
    [-<-=.6] (6,3) – (4,0);
    [->-=.6] (-6,3) – (-4,0);
    [fill=white] (-7,-7) rectangle (-6,-2);
    [fill=white] (-7,7) rectangle (-6,2);
    [fill=white] (7,-7) rectangle (6,-2);
    [fill=white] (7,7) rectangle (6,2);
     at (2,6) [above]i-1;
     at (-4,6) [above]j-1;
     at (-7,-7) [left]i;
     at (-7,7) [left]j;
     at (7,-7) [right]i;
     at (7,7) [right]j;
     
    =
    [baseline=-.6ex, scale=.1] (-8,5) – (-11,5);
     (-8,-5) – (-11,-5);
     (8,5) – (11,5);
     (8,-5) – (11,-5);
     (0,2) to[out=east, in=west] (8,6);
     (0,0) – (6,0);
    [->-=.7, white, double=black, double distance=0.4pt, ultra thick] (6,0) – (8,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-8,-6) – (-5,-6) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-8,6) to[out=east, in=west] (0,2);
    [-<-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,0) – (0,0);
    [->-=.2, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (5,-6) – (8,-6);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-8,-3) – (-6,0);
    [-<-=.6] (8,3) – (6,0);
    [->-=.6] (-8,3) – (-6,0);
    [fill=white] (-9,-7) rectangle (-8,-2);
    [fill=white] (-9,7) rectangle (-8,2);
    [fill=white] (9,-7) rectangle (8,-2);
    [fill=white] (9,7) rectangle (8,2);
    [fill=white] (-5,-7) rectangle (-4,1);
    [fill=white] (5,-7) rectangle (4,1);
     at (2,6) [above]i-1;
     at (-5,6) [above]j-1;
     at (-9,-7) [left]i;
     at (-9,7) [left]j;
     at (9,-7) [right]i;
     at (9,7) [right]j;
     at (0,0) [below]1;
    [opacity=.2, magenta] (-2,0) circle[radius=1];
    [opacity=.2, magenta] (2,0) circle[radius=1];
     
    =(-q^1/6)^2(i-1)[baseline=-.6ex, scale=.1] (-8,5) – (-11,5);
     (-8,-5) – (-11,-5);
     (8,5) – (11,5);
     (8,-5) – (11,-5);
     (0,2) to[out=east, in=west] (8,6);
    [-<-=.5] (-4,-5) – (4,-5);
     (-6,0) – (-4,0);
     (6,0) – (4,0);
    [->-=.5] (-8,-6) – (-4,-6);
    [-<-=.5] (8,-6) – (4,-6);
    [->-=.7, white, double=black, double distance=0.4pt, ultra thick] (6,0) – (8,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-4,-1) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-8,6) to[out=east, in=west] (0,2);
    [->-=.3, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-1);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-8,-3) – (-6,0);
    [-<-=.6] (8,3) – (6,0);
    [->-=.6] (-8,3) – (-6,0);
    [fill=white] (-9,-7) rectangle (-8,-2);
    [fill=white] (-9,7) rectangle (-8,2);
    [fill=white] (9,-7) rectangle (8,-2);
    [fill=white] (9,7) rectangle (8,2);
    [fill=white] (-5,-7) rectangle (-4,1);
    [fill=white] (5,-7) rectangle (4,1);
     at (2,6) [above]i-1;
     at (-5,6) [above]j-1;
     at (-9,-7) [left]i;
     at (-9,7) [left]j;
     at (9,-7) [right]i;
     at (9,7) [right]j;
     at (-6,-6) [below]i-1;
     at (6,-6) [below]i-1;
     at (0,-5) [below]1;
     .




Apply Lemma <ref> to σ_d(k,l).
We have to confirm the web in the second term is equal to σ_d(k,l+1). 
It is deformed as follows.

    [baseline=-.6ex, scale=.1, yshift=2cm] (-19,4) – (-17,4);
     (-19,-8) – (-17,-8);
     (19,4) – (17,4);
     (19,-8) – (17,-8);
    [-<-=.5] (-9,5) – (-16,5);
    [->-=.5] (9,5) – (16,5);
    [->-=.5] (-9,-2) – (-13,-2);
    [->-=.5] (-16,2) to[out=east, in=north west] (-13,-2);
    [->-=.5] (-16,-6) to[out=east, in=south west] (-13,-2);
    [fill=white] (-12,-4) – (-12,0) – (-15,-2) – cycle;
    [->-=.5] (9,-2) – (13,-2);
    [->-=.5] (16,2) to[out=west, in=north east] (13,-2);
    [->-=.5] (16,-6) to[out=west, in=south east] (13,-2);
    [fill=white] (12,-4) – (12,0) – (15,-2) – cycle;
    [->-=.5] (-16,-9) – (-8,-9);
    [-<-=.5] (16,-9) – (8,-9);
    
     (0,2) to[out=east, in=west] (8,6);
    [-<-=.5] (-4,-5) – (4,-5);
    [->-=.5] (-8,-6) – (-4,-6);
    [-<-=.5] (8,-6) – (4,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (-4,-1) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-8,6) to[out=east, in=west] (0,2);
    [->-=.3, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-1);
     (-7,0) – (-4,0);
     (7,0) – (4,0);
    [-<-=.7] (8,-3) – (7,-3) – (7,0);
    [->-=.7] (-8,-3) – (-7,-3) – (-7,0);
    [-<-=.7] (8,3) – (7,3) – (7,0);
    [->-=.7] (-8,3) – (-7,3) – (-7,0);
    [->-=.5] (-8,-8) – (8,-8);
    [-<-=.5] (-8,-10) – (8,-10);
    [fill=white] (-9,-11) rectangle (-8,0);
    [fill=white] (-9,7) rectangle (-8,2);
    [fill=white] (9,-11) rectangle (8,0);
    [fill=white] (9,7) rectangle (8,2);
    [fill=white] (-5,-7) rectangle (-4,1);
    [fill=white] (5,-7) rectangle (4,1);
     at (-2,-6) [above]1;
     at (2,-9) [above]k;
     at (-2,-9) [below]l;
    
    [fill=white] (-17,-11) rectangle (-16,-4);
    [fill=white] (-17,7) rectangle (-16,0);
    [fill=white] (17,-11) rectangle (16,-4);
    [fill=white] (17,7) rectangle (16,0);
        =
    [baseline=-.6ex, scale=.1, yshift=2cm] (-19,4) – (-17,4);
     (-19,-8) – (-17,-8);
     (19,4) – (17,4);
     (19,-8) – (17,-8);
    [->-=.5] (9,5) – (16,5);
    [->-=.7] (-9,-4) – (-9,-2) – (-13,-2);
    [->-=.5] (-16,2) to[out=east, in=north west] (-13,-2);
    [->-=.5] (-16,-6) to[out=east, in=south west] (-13,-2);
    [fill=white] (-12,-4) – (-12,0) – (-15,-2) – cycle;
    [->-=.5] (9,-2) – (13,-2);
    [->-=.5] (16,2) to[out=west, in=north east] (13,-2);
    [->-=.5] (16,-6) to[out=west, in=south east] (13,-2);
    [fill=white] (12,-4) – (12,0) – (15,-2) – cycle;
    [->-=.5] (-16,-9) to[out=east, in=west] (-10,-7);
    [-<-=.5] (16,-9) – (8,-9);
    
     (0,2) to[out=east, in=west] (8,6);
    [-<-=.5] (-4,-5) – (4,-5);
    [->-=.5] (-8,-6) – (-4,-6);
    [-<-=.5] (8,-6) – (4,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (-4,-1) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-16,6) – (-8,6) to[out=east, in=west] (0,2);
    [->-=.3, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-1);
     (-7,0) – (-4,0);
     (7,0) – (4,0);
    [-<-=.7] (8,-3) – (7,-3) – (7,0);
    [->-=.7] (-8,-5) – (-7,-5) – (-7,0);
    [-<-=.7] (8,3) – (7,3) – (7,0);
    [->-=.5] (-16,4) – (-7,4) – (-7,0);
    [->-=.5] (-8,-8) – (8,-8);
    [-<-=.3] (-9,-9) – (-9,-10) – (8,-10);
    [fill=white] (9,-11) rectangle (8,0);
    [fill=white] (9,7) rectangle (8,2);
    [fill=white] (-5,-7) rectangle (-4,1);
    [fill=white] (5,-7) rectangle (4,1);
     at (-2,-6) [above]1;
     at (2,-9) [above]k;
     at (-2,-9) [below]l;
    
    [fill=white] (-10,-4) rectangle (-8,-9);
     (-10,-4) – (-8,-9);
    [fill=white] (-17,-11) rectangle (-16,-4);
    [fill=white] (-17,7) rectangle (-16,0);
    [fill=white] (17,-11) rectangle (16,-4);
    [fill=white] (17,7) rectangle (16,0);
     
    =
    [baseline=-.6ex, scale=.1, yshift=2cm] (-19,4) – (-17,4);
     (-19,-8) – (-17,-8);
     (19,4) – (17,4);
     (19,-8) – (17,-8);
    [->-=.5] (-16,4) – (-10,4);
    [-<-=.5] (-10,4) to[out=east, in=west] (-4,0);
    [->-=.5] (9,5) – (16,5);
     (-8,-5) – (-8,-2) – (-13,-2);
    [->-=.5] (-16,2) to[out=east, in=north west] (-13,-2);
    [->-=.5] (-16,-6) to[out=east, in=south west] (-13,-2);
    [fill=white] (-12,-4) – (-12,0) – (-15,-2) – cycle;
    [->-=.5] (9,-2) – (13,-2);
    [->-=.5] (16,2) to[out=west, in=north east] (13,-2);
    [->-=.5] (16,-6) to[out=west, in=south east] (13,-2);
    [fill=white] (12,-4) – (12,0) – (15,-2) – cycle;
    [->-=.2, ->-=.9] (-16,-8) – (-10,-5) – (-10,4);
    [->-=.5] (-16,-10) to[out=east, in=west] (-9,-8);
    [-<-=.5] (16,-9) – (8,-9);
    
     (0,2) to[out=east, in=west] (8,6);
    [-<-=.5] (-4,-5) – (4,-5);
    [->-=.5] (-7,-6) – (-4,-6);
    [-<-=.5] (8,-6) – (4,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (-4,-1) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-16,6) – (-8,6) to[out=east, in=west] (0,2);
    [->-=.3, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-1);
     (7,0) – (4,0);
    [-<-=.7] (8,-3) – (7,-3) – (7,0);
    [-<-=.7] (8,3) – (7,3) – (7,0);
    [->-=.5] (-8,-8) – (8,-8);
    [-<-=.3] (-8,-9) – (-8,-10) – (8,-10);
    [fill=white] (9,-11) rectangle (8,0);
    [fill=white] (9,7) rectangle (8,2);
    [fill=white] (-5,-7) rectangle (-4,1);
    [fill=white] (5,-7) rectangle (4,1);
     at (-2,-6) [above]1;
     at (2,-9) [above]k;
     at (-2,-9) [below]l;
    
    [fill=white] (-11,-4) rectangle (-9,0);
    [fill=white] (-11,-4) – (-9,0);
    [fill=white] (-9,-5) rectangle (-7,-9);
     (-9,-5) – (-7,-9);
    [fill=white] (-17,-11) rectangle (-16,-4);
    [fill=white] (-17,7) rectangle (-16,0);
    [fill=white] (17,-11) rectangle (16,-4);
    [fill=white] (17,7) rectangle (16,0);
     
       =[baseline=-.6ex, scale=.1, yshift=2cm] (-19,4) – (-17,4);
     (-19,-8) – (-17,-8);
     (19,4) – (17,4);
     (19,-8) – (17,-8);
    [-<-=.5] (-12,0) – (-4,0);
    [->-=.5] (9,5) – (16,5);
    [->-=.3] (-8,-5) – (-8,-2) – (-13,-2);
    [->-=.5] (-16,2) to[out=east, in=north west] (-13,-2);
    [->-=.5] (-16,-6) to[out=east, in=south west] (-13,-2);
    [fill=white] (-12,-5) – (-12,1) – (-16,-2) – cycle;
    [->-=.5] (9,-2) – (13,-2);
    [->-=.5] (16,2) to[out=west, in=north east] (13,-2);
    [->-=.5] (16,-6) to[out=west, in=south east] (13,-2);
    [fill=white] (12,-4) – (12,0) – (15,-2) – cycle;
    [->-=.5] (-16,-9) to[out=east, in=west] (-9,-7);
    [-<-=.5] (16,-9) – (8,-9);
    
     (0,2) to[out=east, in=west] (8,6);
    [-<-=.5] (-4,-5) – (4,-5);
    [->-=.5] (-7,-6) – (-4,-6);
    [-<-=.5] (8,-6) – (4,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (-4,-1) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-16,6) – (-8,6) to[out=east, in=west] (0,2);
    [->-=.3, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-1);
     (7,0) – (4,0);
    [-<-=.7] (8,-3) – (7,-3) – (7,0);
    [-<-=.7] (8,3) – (7,3) – (7,0);
    [->-=.5] (-8,-8) – (8,-8);
    [-<-=.3] (-8,-9) – (-8,-10) – (8,-10);
    [fill=white] (9,-11) rectangle (8,0);
    [fill=white] (9,7) rectangle (8,2);
    [fill=white] (-5,-7) rectangle (-4,1);
    [fill=white] (5,-7) rectangle (4,1);
     at (-2,-6) [above]1;
     at (2,-9) [above]k;
     at (-2,-9) [below]l;
    
    [fill=white] (-9,-5) rectangle (-7,-9);
     (-9,-5) – (-7,-9);
    [fill=white] (-17,-11) rectangle (-16,-4);
    [fill=white] (-17,7) rectangle (-16,0);
    [fill=white] (17,-11) rectangle (16,-4);
    [fill=white] (17,7) rectangle (16,0);
     
    =[baseline=-.6ex, scale=.1, yshift=2cm] (-19,4) – (-17,4);
     (-19,-8) – (-17,-8);
     (19,4) – (17,4);
     (19,-8) – (17,-8);
    [-<-=.5] (-10,0) – (-4,0);
    [->-=.5] (9,5) – (16,5);
     (-12,-2) – (-11,-2);
    [->-=.5] (-8,-5) – (-8,-3) – (-10,-3);
    [->-=.5] (-16,2) to[out=east, in=north west] (-13,-2);
    [->-=.5] (-16,-6) to[out=east, in=south west] (-13,-2);
    [fill=white] (-12,-5) – (-12,1) – (-16,-2) – cycle;
    [->-=.5] (9,-2) – (13,-2);
    [->-=.5] (16,2) to[out=west, in=north east] (13,-2);
    [->-=.5] (16,-6) to[out=west, in=south east] (13,-2);
    [fill=white] (12,-4) – (12,0) – (15,-2) – cycle;
    [->-=.5] (-16,-9) to[out=east, in=west] (-9,-7);
    [-<-=.5] (16,-9) – (8,-9);
    
     (0,2) to[out=east, in=west] (8,6);
    [-<-=.5] (-4,-5) – (4,-5);
    [->-=.5] (-7,-6) – (-4,-6);
    [-<-=.5] (8,-6) – (4,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (-4,-1) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-16,6) – (-8,6) to[out=east, in=west] (0,2);
    [->-=.3, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-1);
     (7,0) – (4,0);
    [-<-=.7] (8,-3) – (7,-3) – (7,0);
    [-<-=.7] (8,3) – (7,3) – (7,0);
    [->-=.5] (-8,-8) – (8,-8);
    [-<-=.3] (-8,-9) – (-8,-10) – (8,-10);
    [fill=white] (9,-11) rectangle (8,0);
    [fill=white] (9,7) rectangle (8,2);
    [fill=white] (-5,-7) rectangle (-4,1);
    [fill=white] (5,-7) rectangle (4,1);
     at (-2,-6) [above]1;
     at (2,-9) [above]k;
     at (-2,-9) [below]l;
    
    [fill=white] (-11,-9) rectangle (-10,1);
    [fill=white] (-9,-5) rectangle (-7,-9);
     (-9,-5) – (-7,-9);
    [fill=white] (-17,-11) rectangle (-16,-4);
    [fill=white] (-17,7) rectangle (-16,0);
    [fill=white] (17,-11) rectangle (16,-4);
    [fill=white] (17,7) rectangle (16,0);
     
       =[baseline=-.6ex, scale=.1, yshift=2cm] (-19,4) – (-17,4);
     (-19,-8) – (-17,-8);
     (19,4) – (17,4);
     (19,-8) – (17,-8);
    [-<-=.5] (-10,0) – (-4,0);
    [->-=.5] (9,5) – (16,5);
     (-11,-2) – (-12,-2);
    [->-=.5] (-16,2) to[out=east, in=north west] (-13,-2);
    [->-=.5] (-16,-6) to[out=east, in=south west] (-13,-2);
    [fill=white] (-12,-5) – (-12,1) – (-16,-2) – cycle;
    [->-=.5] (9,-2) – (13,-2);
    [->-=.5] (16,2) to[out=west, in=north east] (13,-2);
    [->-=.5] (16,-6) to[out=west, in=south east] (13,-2);
    [fill=white] (12,-4) – (12,0) – (15,-2) – cycle;
    [->-=.5] (-16,-9) to[out=east, in=west] (-10,-7);
    [-<-=.5] (16,-9) – (8,-9);
    
     (0,2) to[out=east, in=west] (8,6);
    [-<-=.5] (-4,-5) – (4,-5);
    [->-=.5] (-10,-6) – (-4,-6);
    [-<-=.5] (8,-6) – (4,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (-4,-1) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-16,6) – (-8,6) to[out=east, in=west] (0,2);
    [->-=.3, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-1);
     (7,0) – (4,0);
    [-<-=.7] (8,-3) – (7,-3) – (7,0);
    [-<-=.7] (8,3) – (7,3) – (7,0);
    [->-=.5] (-10,-8) – (8,-8);
    [-<-=.3] (-10,-10) – (8,-10);
    [fill=white] (9,-11) rectangle (8,0);
    [fill=white] (9,7) rectangle (8,2);
    [fill=white] (-5,-7) rectangle (-4,1);
    [fill=white] (5,-7) rectangle (4,1);
     at (-2,-6) [above]1;
     at (2,-9) [above]k;
     at (-2,-9) [below]l;
    
    [fill=white] (-11,-11) rectangle (-10,1);
    [fill=white] (-17,-11) rectangle (-16,-4);
    [fill=white] (-17,7) rectangle (-16,0);
    [fill=white] (17,-11) rectangle (16,-4);
    [fill=white] (17,7) rectangle (16,0);
     
    =[baseline=-.6ex, scale=.1, yshift=2cm] (-19,4) – (-17,4);
     (-19,-8) – (-17,-8);
     (19,4) – (17,4);
     (19,-8) – (17,-8);
    [->-=.5] (9,5) – (16,5);
     (-11,-2) – (-12,-2);
    [->-=.5] (-16,2) to[out=east, in=north west] (-13,-2);
    [->-=.5] (-16,-6) to[out=east, in=south west] (-13,-2);
    [fill=white] (-12,-5) – (-12,1) – (-16,-2) – cycle;
    [->-=.5] (9,-2) – (13,-2);
    [->-=.5] (16,2) to[out=west, in=north east] (13,-2);
    [->-=.5] (16,-6) to[out=west, in=south east] (13,-2);
    [fill=white] (12,-4) – (12,0) – (15,-2) – cycle;
    [->-=.5] (-16,-9) to[out=east, in=west] (-10,-7);
    [-<-=.5] (16,-9) – (8,-9);
    
     (0,2) to[out=east, in=west] (8,6);
    [-<-=.5] (-10,-5) – (4,-5);
    [-<-=.5] (8,-6) – (4,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (-10,-1) – (-4,-1) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-16,6) – (-8,6) to[out=east, in=west] (0,2);
    [->-=.3, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-1);
     (7,0) – (4,0);
    [-<-=.7] (8,-3) – (7,-3) – (7,0);
    [-<-=.7] (8,3) – (7,3) – (7,0);
    [->-=.5] (-10,-8) – (8,-8);
    [-<-=.3] (-10,-10) – (8,-10);
    [fill=white] (9,-11) rectangle (8,0);
    [fill=white] (9,7) rectangle (8,2);
    [fill=white] (5,-7) rectangle (4,1);
     at (-2,-6) [above]1;
     at (2,-9) [above]k;
     at (-2,-9) [below]l;
    
    [fill=white] (-11,-11) rectangle (-10,1);
    [fill=white] (-17,-11) rectangle (-16,-4);
    [fill=white] (-17,7) rectangle (-16,0);
    [fill=white] (17,-11) rectangle (16,-4);
    [fill=white] (17,7) rectangle (16,0);
     .

In the first equation, 
we remove an A_2 clasp by using its definition and property of the triangle in Lemma <ref>.
One can also apply the same deformation on the right side of the web and obtain the following equation:

    [baseline=-.6ex, scale=.1, yshift=2cm] (-19,4) – (-17,4);
     (-19,-8) – (-17,-8);
     (19,4) – (17,4);
     (19,-8) – (17,-8);
    [-<-=.5] (-9,5) – (-16,5);
    [->-=.5] (9,5) – (16,5);
    [->-=.5] (-9,-2) – (-13,-2);
    [->-=.5] (-16,2) to[out=east, in=north west] (-13,-2);
    [->-=.5] (-16,-6) to[out=east, in=south west] (-13,-2);
    [fill=white] (-12,-4) – (-12,0) – (-15,-2) – cycle;
    [->-=.5] (9,-2) – (13,-2);
    [->-=.5] (16,2) to[out=west, in=north east] (13,-2);
    [->-=.5] (16,-6) to[out=west, in=south east] (13,-2);
    [fill=white] (12,-4) – (12,0) – (15,-2) – cycle;
    [->-=.5] (-16,-9) – (-8,-9);
    [-<-=.5] (16,-9) – (8,-9);
    
     (0,2) to[out=east, in=west] (8,6);
    [-<-=.5] (-4,-5) – (4,-5);
    [->-=.5] (-8,-6) – (-4,-6);
    [-<-=.5] (8,-6) – (4,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (-4,-1) to[out=east, in=west] (0,6);
    [->-=.5, white, double=black, double distance=0.4pt, ultra thick] (-8,6) to[out=east, in=west] (0,2);
    [->-=.3, white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (4,-1);
     (-7,0) – (-4,0);
     (7,0) – (4,0);
    [-<-=.7] (8,-3) – (7,-3) – (7,0);
    [->-=.7] (-8,-3) – (-7,-3) – (-7,0);
    [-<-=.7] (8,3) – (7,3) – (7,0);
    [->-=.7] (-8,3) – (-7,3) – (-7,0);
    [->-=.5] (-8,-8) – (8,-8);
    [-<-=.5] (-8,-10) – (8,-10);
    [fill=white] (-9,-11) rectangle (-8,0);
    [fill=white] (-9,7) rectangle (-8,2);
    [fill=white] (9,-11) rectangle (8,0);
    [fill=white] (9,7) rectangle (8,2);
    [fill=white] (-5,-7) rectangle (-4,1);
    [fill=white] (5,-7) rectangle (4,1);
     at (-2,-6) [above]1;
     at (2,-9) [above]k;
     at (-2,-9) [below]l;
    
    [fill=white] (-17,-11) rectangle (-16,-4);
    [fill=white] (-17,7) rectangle (-16,0);
    [fill=white] (17,-11) rectangle (16,-4);
    [fill=white] (17,7) rectangle (16,0);
     
    =[baseline=-.6ex, scale=.1] (-17,5) – (-14,5);
     (-17,-5) – (-14,-5);
     (17,5) – (14,5);
     (17,-5) – (14,-5);
    [->-=.5] (-14,6) – (-6,6);
    [->-=.5] (-14,-6) – (-6,-6);
    [->-=.5] (-6,0) – (-11,0);
    [->-=.5] (-14,4) to[out=east, in=north east] (-11,0);
    [->-=.5] (-14,-4) to[out=east, in=south east] (-11,0);
    [fill=white] (-12,0) – (-9,2) – (-9,-2) – cycle;
    [-<-=.5] (14,6) – (6,6);
    [-<-=.5] (14,-6) – (6,-6);
    [-<-=.5] (6,0) – (11,0);
    [-<-=.5] (14,4) to[out=west, in=north west] (11,0);
    [-<-=.5] (14,-4) to[out=west, in=south west] (11,0);
    [fill=white] (12,0) – (9,2) – (9,-2) – cycle;
    
    [-<-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,-4) – (6,-4);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,-1) to[out=east, in=west] (0,6);
    [->-=.9, white, double=black, double distance=0.4pt, ultra thick] (-6,6) to[out=east, in=west] (0,-1);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-1) to[out=east, in=west] (6,6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=west] (6,-1);
    [fill=white] (-7,8) rectangle (-6,4);
    [fill=white] (-7,-8) rectangle (-6,2);
    [fill=white] (7,8) rectangle (6,4);
    [fill=white] (7,-8) rectangle (6,2);
     at (3,-5)[above]k;
     at (3,-5)[below]l+1;
    
    [fill=white] (-14,-8) rectangle (-15,-2);
    [fill=white] (-14,8) rectangle (-15,2);
    [fill=white] (14,-8) rectangle (15,-2);
    [fill=white] (14,8) rectangle (15,2);
     
    =σ_d(k,l+1).




[parallel m-full twist formula]
Let k=(k_1,…,k_m) be an m-tuple of integers, k_0=d=min{s,t}, and δ=| s-t |.

    [baseline=-.6ex][xshift=-1cm]
     
    (-.5,.4) – +(-.2,0)
    (-.5,-.4) – +(-.2,0);
    [->-=1, white, double=black, double distance=0.4pt, ultra thick] 
    (-.5,-.4) to[out=east, in=west] (.0,.4);
    [->-=1, white, double=black, double distance=0.4pt, ultra thick] 
    (-.5,.4) to[out=east, in=west] (.0,-.4);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (0,-.4) to[out=east, in=west] (.5,.4);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (0,.4) to[out=east, in=west] (.5,-.4);
    [fill=white] (-.5,-.6) rectangle +(-.1,.4);
    [fill=white] (-.5,.6) rectangle +(-.1,-.4);
     at (-.5,-.6)[left]s;
     at (-.5,.6)[left]t;
     at (.0,.0)⋯;
     at (.0,-.4)[below]m full twists;
    [xshift=1cm]
    
    (.5,-.4) – +(.2,0)
    (.5,.4) – +(.2,0);
    [->-=1, white, double=black, double distance=0.4pt, ultra thick] 
    (-.5,-.4) to[out=east, in=west] (.0,.4);
    [->-=1, white, double=black, double distance=0.4pt, ultra thick] 
    (-.5,.4) to[out=east, in=west] (.0,-.4);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (0,-.4) to[out=east, in=west] (.5,.4);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (0,.4) to[out=east, in=west] (.5,-.4);
    [fill=white] (.5,-.6) rectangle +(.1,.4);
    [fill=white] (.5,.6) rectangle +(.1,-.4);
     at (.5,-.6)[right]s;
     at (.5,.6)[right]t;
        =q^-m/3k_0(k_0+δ)-mk_0∑_k_0≥ k_1≥⋯≥ k_m≥ 0D(k)
     [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (-3,0) – (0,3) – (3,0) – (0,-3) – cycle;
     (0,3) – (0,-3);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,6)[above]t-d+k_m;
     at (0,-6)[below]s-d+k_m;
     at (-3,4)[below left]d-k_m;
     at (-3,-4)[above left]d-k_m;
     at (3,4)[below right]d-k_m;
     at (3,-4)[above right]d-k_m;
     ,

where 

    D(k)=q^∑_i=1^mk_i(k_i+δ)+k_iq^1/2(k_0-k_m)(q)_k_0+δ/(q)_k_m+δd  k_1',k_2',…,k_m',k_m_q

and k_i+1'=k_i-k_i+1 for i=0,1,…,m-1.


For a positive integer m, 
we denote a right-handed m half twist by

    [baseline=-.6ex, scale=.1] (-5,2) – (5,2);
     (-5,-2) – (5,-2);
    [fill=white] (-3,-3) rectangle (3,3);
     at (0,0)m;
     
    =
     [baseline=-.6ex, scale=.1][xshift=-5cm]
     (-5,2) – +(-2,0);
     (-5,-2) – +(-2,0);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (-5,-2) to[out=east, in=west] (0,2);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (-5,2) to[out=east, in=west] (0,-2);
     at (0,0)⋯;
     at (0,-2)[below]m crossings;
    [xshift=5cm]
    
    (5,-2) – +(2,0)
    (5,2) – +(2,0);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (0,-2) to[out=east, in=west] (5,2);
    [white, double=black, double distance=0.4pt, ultra thick] 
    (0,2) to[out=east, in=west] (5,-2);



By sliding the right-hand triangle,

    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (-3,0) – (0,3) – (3,0) – (0,-3) – cycle;
     (0,3) – (0,-3);
    [xshift=16cm]
    [white, double=black, double distance=0.4pt, ultra thick] (-6,-5) to[out=east, in=west] (0,5);
    [white, double=black, double distance=0.4pt, ultra thick] (-6,5) to[out=east, in=west] (0,-5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-5) to[out=east, in=west] (6,5);
    [white, double=black, double distance=0.4pt, ultra thick] (0,5) to[out=east, in=west] (6,-5);
    [->-=.7] (-9,5) – (-6,5);
    [->-=.7] (-9,-5) – (-6,-5);
    [->-=.5] (6,5) – (9,5);
    [->-=.5] (6,-5) – (9,-5);
     (10,5) – (12,5);
     (10,-5) – (12,-5);
    [fill=white] (9,-7) rectangle (10,-3);
    [fill=white] (9,7) rectangle (10,3);
     at (10,-7)[right]s;
     at (10,7)[right]t;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,6)[above]k+(t-d);
     at (0,-6)[below]k+(s-d);
     at (-3,4)[below left]d-k;
     at (-3,-4)[above left]d-k;
     at (3,4)[below right]d-k;
     at (3,-4)[above right]d-k;
     
       =((-1)^d-kq^-1/6(d-k)^2-1/2(d-k)q^-1/3(d-k)(k+s-d)q^-1/3(d-k)(k+t-d))^2
         × [baseline=-.6ex, scale=.1][xshift=0cm]
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) to[out=east, in=south] (-2,0) to[out=north, in=west](0,6);
    [-<-=.2, white, double=black, double distance=0.4pt, ultra thick] (-9,0) – (0,0);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,6) to[out=east, in=west] (0,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-6) to[out=east, in=west] (9,6);
    [-<-=.8, white, double=black, double distance=0.4pt, ultra thick] (0,0) – (9,0);
    [white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=north] (2,0) to[out=south, in=west] (9,-6);
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (0,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]d-k;
     at (-3,-4)[above left]d-k;
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (0,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]s;
     at (7,7)[right]t;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]d-k;
     at (3,-4)[above right]d-k;
     at (0,6)[above]k+(s-d);
     at (0,-6)[below]k+(t-d);

and

    [baseline=-.6ex, scale=.1][xshift=0cm]
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) to[out=east, in=south] (-2,0) to[out=north, in=west](0,6);
    [-<-=.2, white, double=black, double distance=0.4pt, ultra thick] (-9,0) – (0,0);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,6) to[out=east, in=west] (0,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-6) to[out=east, in=west] (9,6);
    [-<-=.8, white, double=black, double distance=0.4pt, ultra thick] (0,0) – (9,0);
    [white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=north] (2,0) to[out=south, in=west] (9,-6);
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (0,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]d-k;
     at (-3,-4)[above left]d-k;
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (0,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]s;
     at (7,7)[right]t;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]d-k;
     at (3,-4)[above right]d-k;
     at (0,6)[above]k+(s-d);
     at (0,-6)[below]k+(t-d);
        = [baseline=-.6ex, scale=.1][xshift=0cm]
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) to[out=east, in=south] (-2,0) to[out=north, in=west](0,6);
    [-<-=.2, white, double=black, double distance=0.4pt, ultra thick] (-9,0) – (0,0);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,6) to[out=east, in=west] (0,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-6) to[out=east, in=west] (9,6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,0) – (9,0);
    [white, double=black, double distance=0.4pt, ultra thick] (0,6) to[out=east, in=north] (2,0) to[out=south, in=west] (9,-6);
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.9] (-6,6) – (0,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]d-k;
     at (-3,-4)[above left]d-k;
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (0,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]s;
     at (7,7)[right]t;
    [fill=white] (-1,-7) rectangle (-2,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]d-k;
     at (3,-4)[above right]d-k;
     at (0,6)[above]k+(s-d);
     at (0,-6)[below]k+(t-d);
     
       =(-q^1/6)^2(k+s-d)(d-k) [baseline=-.6ex, scale=.1][xshift=0cm]
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) to[out=east, in=west](0,-2);
    [-<-=.2, white, double=black, double distance=0.4pt, ultra thick] (-9,0) – (0,0);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,6) to[out=east, in=west] (0,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-6) to[out=east, in=west] (9,6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,0) – (9,0);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-2) to[out=east, in=west] (9,-6);
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (0,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]d-k;
     at (-3,-4)[above left]d-k;
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (0,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]s;
     at (7,7)[right]t;
    [fill=white] (-1,-7) rectangle (-2,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]d-k;
     at (3,-4)[above right]d-k;
     at (-9,6)[above]k+(t-d);
     at (-9,-6)[below]k+(s-d);
     
       =q^1/3(d-k)(k+s-d) [baseline=-.6ex, scale=.1][xshift=0cm]
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) to[out=east, in=west](0,-2);
    [-<-=.9, white, double=black, double distance=0.4pt, ultra thick] (-9,0) – (0,0);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,6) to[out=east, in=west] (0,-6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-6) to[out=east, in=west] (9,6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,0) – (9,0);
    [white, double=black, double distance=0.4pt, ultra thick] (0,-2) to[out=east, in=west] (9,-6);
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (0,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]d-k;
     at (-3,-4)[above left]d-k;
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (0,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]s;
     at (7,7)[right]t;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]d-k;
     at (3,-4)[above right]d-k;
     at (-9,6)[above]k+(t-d);
     at (-9,-6)[below]k+(s-d);
     .

Let m be positive integers and {k_i}_i are non-negative integers satisfying k_i≥ k_i+1 for 0≤ i≤ m-1. 
We set m_i=(m-i), k_0=d, t_i=k_i+(t-d), and s_i=k_i+(s-d). 
Then, as the above calculation, we obtain

    [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (-3,0) – (0,3) – (3,0) – (0,-3) – cycle;
     (0,3) – (0,-3);
    [->-=.5] (7,5) – (12,5);
    [->-=.5] (7,-5) – (12,-5);
    [->-=.5] (20,5) – (25,5);
    [->-=.5] (20,-5) – (25,-5);
     (26,5) – (28,5);
     (26,-5) – (28,-5);
    [fill=white] (12,-6) rectangle (20,6);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
    [fill=white] (25,-7) rectangle (26,-3);
    [fill=white] (25,7) rectangle (26,3);
     at (16,0)2m_i;
     at (-7,-7)[left]s_i-1;
     at (-7,7)[left]t_i-1;
     at (7,-7)[right]s_i-1;
     at (7,7)[right]t_i-1;
     at (26,-7)[right]s_i-1;
     at (26,7)[right]t_i-1;
     at (0,6)[above]t_i;
     at (0,-6)[below]s_i;
     at (-3,4)[below left]k_i-1-k_i;
     at (-3,-4)[above left]k_i-1-k_i;
     
         =q^-m_i/3(k_i-1-k_i)^2-m_i(k_i-1-k_i)q^-m_i/3(k_i-1-k_i)(k_i+s-d)q^-m_i-1/3(k_i-1-k_i)(k_i+t-d)
           × q^-2/3(k_i-1-k_i)(k_i+t-d) [baseline=-.6ex, scale=.1][xshift=0cm]
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) to[out=east, in=west](-4,-6);
    [-<-=.9, white, double=black, double distance=0.4pt, ultra thick] (-9,0) – (0,0);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,6) to[out=east, in=west] (-4,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (4,-3) to[out=east, in=west] (9,6);
    [white, double=black, double distance=0.4pt, ultra thick] (0,0) – (9,0);
    [white, double=black, double distance=0.4pt, ultra thick] (4,-6) to[out=east, in=west] (9,-6);
    [fill=white] (-4,-7) rectangle (4,-2);
     at (0,-5)2m_i;
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (0,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]k_i-1-k_i;
     at (-3,-4)[above left]k_i-1-k_i;
     at (-7,-7)[left]s_i-1;
     at (-7,7)[left]t_i-1;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (0,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]s_i-1;
     at (7,7)[right]t_i-1;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]k_i-1-k_i;
     at (3,-4)[above right]k_i-1-k_i;
     at (-11,6)[above]t_i;
     at (-9,-6)[below]s_i;
     
         =q^-m_i/3(k_i-1-k_i)^2-m_i(k_i-1-k_i)q^-m_i/3(k_i-1-k_i)(k_i+s-d)q^-m_i/3(k_i-1-k_i)(k_i+t-d)
           × [baseline=-.6ex, scale=.1][xshift=0cm]
     (-9,0) – (-8,0);
     (9,0) – (8,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-7,4) – (7,4);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-7,-3) – (-4,-3);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (4,-3) – (7,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) – (9,-6);
    [fill=white] (-4,-7) rectangle (4,-2);
    [fill=white] (-8,-4) rectangle (-7,7);
    [fill=white] (8,-4) rectangle (7,7);
     at (0,-5) 2m_i;
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (1,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]k_i-1-k_i;
     at (-3,-4)[above left]k_i-1-k_i;
     at (-7,-7)[left]s_i-1;
     at (-7,7)[left]t_i-1;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-1,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]s_i-1;
     at (7,7)[right]t_i-1;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]k_i-1-k_i;
     at (3,-4)[above right]k_i-1-k_i;
     at (-11,6)[above]t_i;
     at (-9,-6)[below]s_i;
     at (0,4)[below]k_i-1-k_i;
     
         =q^-m_i/3(k_i-1-k_i)^2-m_i(k_i-1-k_i)q^-m_i/3(k_i-1-k_i)k_iq^-m_i/3(k_i-1-k_i)(k_i+δ)
           × [baseline=-.6ex, scale=.1][xshift=0cm]
     (-9,0) – (-8,0);
     (9,0) – (8,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-7,4) – (7,4);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-7,-3) – (-4,-3);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (4,-3) – (7,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) – (9,-6);
    [fill=white] (-4,-7) rectangle (4,-2);
    [fill=white] (-8,-4) rectangle (-7,7);
    [fill=white] (8,-4) rectangle (7,7);
     at (0,-5) 2m_i;
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (1,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]k_i-1-k_i;
     at (-3,-4)[above left]k_i-1-k_i;
     at (-7,-7)[left]t_i-1;
     at (-7,7)[left]t_i-1;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-1,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]t_i-1;
     at (7,7)[right]t_i-1;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]k_i-1-k_i;
     at (3,-4)[above right]k_i-1-k_i;
     at (-11,6)[above]t_i;
     at (-9,-6)[below]s_i;
     at (0,4)[below]k_i-1-k_i;
     
         =q^-m_i/3k_i-1(k_i-1+δ)-m_ik_i-1q^m_i/3k_i(k_i+δ)+m_ik_i [baseline=-.6ex, scale=.1][xshift=0cm]
     (-9,0) – (-8,0);
     (9,0) – (8,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-7,4) – (7,4);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-7,-3) – (-4,-3);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (4,-3) – (7,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) – (9,-6);
    [fill=white] (-4,-7) rectangle (4,-2);
    [fill=white] (-8,-4) rectangle (-7,7);
    [fill=white] (8,-4) rectangle (7,7);
     at (0,-5) 2m_i;
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (1,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]k_i-1-k_i;
     at (-3,-4)[above left]k_i-1-k_i;
     at (-7,-7)[left]t_i-1;
     at (-7,7)[left]t_i-1;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-1,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]t_i-1;
     at (7,7)[right]t_i-1;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]k_i-1-k_i;
     at (3,-4)[above right]k_i-1-k_i;
     at (-11,6)[above]t_i;
     at (-9,-6)[below]s_i;
     at (0,4)[below]k_i-1-k_i;
     .

Let τ_0 be the parallel m full twist in the LHS of Theorem <ref> and

    τ_i
    = [baseline=-.6ex, scale=.1][xshift=0cm]
     (-9,0) – (-8,0);
     (9,0) – (8,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-7,4) – (7,4);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-7,-3) – (-4,-3);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (4,-3) – (7,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) – (9,-6);
    [fill=white] (-4,-7) rectangle (4,-2);
    [fill=white] (-8,-4) rectangle (-7,7);
    [fill=white] (8,-4) rectangle (7,7);
     at (0,-5) 2m_i;
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (1,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]k_i-1-k_i;
     at (-3,-4)[above left]k_i-1-k_i;
     at (-7,-7)[left]s_i-1;
     at (-7,7)[left]t_i-1;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-1,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]s_i-1;
     at (7,7)[right]t_i-1;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]k_i-1-k_i;
     at (3,-4)[above right]k_i-1-k_i;
     at (-11,6)[above]t_i;
     at (-9,-6)[below]s_i;
     at (0,4)[below]k_i-1-k_i;

for i=1,2,…,m-1.
Then,
we apply Theorem <ref>, the index of summation is k_i+1, to the left-most full twist in the box
[baseline=-.6ex, scale=.1][draw, rectangle] 2m_i; 
of τ_i and the above deformation.

    τ_i   =q^-1/3k_i(k_i+δ)-k_i/2∑_k_i+1=0^k_i
    q^k_i+1(k_i+1+δ)+k_i+1/2(q)_k_i+δ/(q)_k_i+1+δk_i k_i+1_q
       × q^-m_i+1/3k_i(k_i+δ)-m_i+1k_iq^m_i+1/3k_i+1(k_i+1+δ)+m_i+1k_i+1 [baseline=-.6ex, scale=.1][xshift=0cm]
     (-9,0) – (-8,0);
     (9,0) – (8,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-7,4) – (7,4);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-7,-3) – (-4,-3);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (4,-3) – (7,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) – (9,-6);
    [fill=lightgray] (-4,-7) rectangle (4,-2);
    [fill=white] (-8,-4) rectangle (-7,7);
    [fill=white] (8,-4) rectangle (7,7);
     at (0,-5) τ_i+1;
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (1,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]k_i-1-k_i;
     at (-3,-4)[above left]k_i-1-k_i;
     at (-7,-7)[left]t_i-1;
     at (-7,7)[left]t_i-1;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-1,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]t_i-1;
     at (7,7)[right]t_i-1;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]k_i-1-k_i;
     at (3,-4)[above right]k_i-1-k_i;
     at (-11,6)[above]t_i;
     at (-9,-6)[below]s_i;
     at (0,4)[below]k_i-1-k_i;
     
       =∑_k_i+1=0^k_i
    q^-m_i/3k_i(k_i+δ)+m_i+1/3k_i+1(k_i+1+δ)
    q^-1/2(k_i-k_i+1)q^-m_i+1(k_i-k_i+1)
    q^k_i+1(k_i+1+δ)
         ×(q)_k_i+δ/(q)_k_i+1+δk_i k_i+1_q [baseline=-.6ex, scale=.1][xshift=0cm]
     (-9,0) – (-8,0);
     (9,0) – (8,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-7,4) – (7,4);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-7,-3) – (-4,-3);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (4,-3) – (7,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) – (9,-6);
    [fill=lightgray] (-4,-7) rectangle (4,-2);
    [fill=white] (-8,-4) rectangle (-7,7);
    [fill=white] (8,-4) rectangle (7,7);
     at (0,-5) τ_i+1;
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (1,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]k_i-1-k_i;
     at (-3,-4)[above left]k_i-1-k_i;
     at (-7,-7)[left]t_i-1;
     at (-7,7)[left]t_i-1;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-1,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]t_i-1;
     at (7,7)[right]t_i-1;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]k_i-1-k_i;
     at (3,-4)[above right]k_i-1-k_i;
     at (-11,6)[above]t_i;
     at (-9,-6)[below]s_i;
     at (0,4)[below]k_i-1-k_i;
     
       =∑_k_i+1=0^k_i
    q^-m_i/3k_i(k_i+δ)+m_i+1/3k_i+1(k_i+1+δ)
    q^-1/2(k_i-k_i+1)q^-m_ik_i+m_i+1k_i+1
         × 
    q^k_iq^k_i+1(k_i+1+δ)(q)_k_i+δ/(q)_k_i+1+δk_i k_i+1_q [baseline=-.6ex, scale=.1][xshift=0cm]
     (-9,0) – (-8,0);
     (9,0) – (8,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-7,4) – (7,4);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-7,-3) – (-4,-3);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (4,-3) – (7,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) – (9,-6);
    [fill=lightgray] (-4,-7) rectangle (4,-2);
    [fill=white] (-8,-4) rectangle (-7,7);
    [fill=white] (8,-4) rectangle (7,7);
     at (0,-5) τ_i+1;
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (1,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]k_i-1-k_i;
     at (-3,-4)[above left]k_i-1-k_i;
     at (-7,-7)[left]t_i-1;
     at (-7,7)[left]t_i-1;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-1,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]t_i-1;
     at (7,7)[right]t_i-1;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]k_i-1-k_i;
     at (3,-4)[above right]k_i-1-k_i;
     at (-11,6)[above]t_i;
     at (-9,-6)[below]s_i;
     at (0,4)[below]k_i-1-k_i;
     ,

where the shaded box with τ_i+1 means replacement of the box with τ_i+1.
We remark that m_0=m, t_0=t, s_0=s, and min{s_i-1,t_i-1}=k_i-1.
By using a similar way to (<ref>), 
we can confirm that 

    [baseline=-.6ex, scale=.1][xshift=0cm]
     (-9,0) – (-8,0);
     (9,0) – (8,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-7,4) – (7,4);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-7,-3) – (-4,-3);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (4,-3) – (7,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) – (9,-6);
    [fill=lightgray] (-4,-7) rectangle (4,-2);
    [fill=white] (-8,-4) rectangle (-7,7);
    [fill=white] (8,-4) rectangle (7,7);
     at (0,-5) τ_i+1;
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (1,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]k_i-1-k_i;
     at (-3,-4)[above left]k_i-1-k_i;
     at (-7,-7)[left]t_i-1;
     at (-7,7)[left]t_i-1;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-1,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]t_i-1;
     at (7,7)[right]t_i-1;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]k_i-1-k_i;
     at (3,-4)[above right]k_i-1-k_i;
     at (-11,6)[above]t_i;
     at (-9,-6)[below]s_i;
     at (0,4)[below]k_i-1-k_i;
     
    = [baseline=-.6ex, scale=.12][xshift=0cm]
     (-9,0) – (-8,0);
     (9,0) – (8,0);
    [-<-=.5, white, double=black, double distance=0.4pt, ultra thick] (-7,4) – (7,4);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (-7,-3) – (-4,-3);
    [->-=.6, white, double=black, double distance=0.4pt, ultra thick] (4,-3) – (7,-3);
    [white, double=black, double distance=0.4pt, ultra thick] (-9,-6) – (9,-6);
    [fill=white] (-4,-7) rectangle (4,-2);
    [fill=white] (-8,-4) rectangle (-7,7);
    [fill=white] (8,-4) rectangle (7,7);
     at (0,-5) 2m_i+1;
    [xshift=-9cm]
     (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
    [->-=.5] (-6,6) – (1,6);
    [->-=.5] (-6,-6) – (0,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [fill=white] (-3,0) – (0,3) – (0,-3) – cycle;
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
     at (-3,4)[below left]k_i-1-k_i+1;
     at (-3,-4)[above left]k_i-1-k_i+1;
     at (-7,-7)[left]s_i-1;
     at (-7,7)[left]t_i-1;
    [xshift=9cm]
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-1,6) – (6,6);
    [->-=.5] (0,-6) – (6,-6);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (3,0) – (0,3) – (0,-3) – cycle;
     at (7,-7)[right]s_i-1;
     at (7,7)[right]t_i-1;
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (3,4)[below right]k_i-1-k_i+1;
     at (3,-4)[above right]k_i-1-k_i+1;
     at (-11,6)[above]t_i+1;
     at (-9,-6)[below]s_i+1;
     at (0,4)[below]k_i-1-k_i+1;
     .

Consequently, 

    τ_0   =∑_k_0≥ k_1≥⋯≥ k_m≥ 0∏_i=0^m-1
    q^-m_i/3k_i(k_i+δ)+m_i+1/3k_i+1(k_i+1+δ)
    q^-1/2(k_i-k_i+1)q^-m_ik_i+m_i+1k_i+1
         × 
    q^k_i+1(k_i+1+δ)+k_i(q)_k_i+δ/(q)_k_i+1+δk_i k_i+1_q [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (-3,0) – (0,3) – (3,0) – (0,-3) – cycle;
     (0,3) – (0,-3);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,6)[above]t-d+k_m;
     at (0,-6)[below]s-d+k_m;
     at (-3,4)[below left]d-k_m;
     at (-3,-4)[above left]d-k_m;
     at (3,4)[below right]d-k_m;
     at (3,-4)[above right]d-k_m;
     
       =∑_k_0≥ k_1≥⋯≥ k_m≥ 0
    q^-m/3k_0(k_0+δ)
    q^-1/2(k_0-k_m)q^-mk_0
    q^∑_i=0^m-1k_i+1(k_i+1+δ)+k_i
         ×(q)_k_0+δ/(q)_k_m+δ(q)_k_0/(q)_k_0-k_1(q)_k_1-k_2… (q)_k_m-1-k_m(q)_k_m [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (-3,0) – (0,3) – (3,0) – (0,-3) – cycle;
     (0,3) – (0,-3);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,6)[above]t-d+k_m;
     at (0,-6)[below]s-d+k_m;
     at (-3,4)[below left]d-k_m;
     at (-3,-4)[above left]d-k_m;
     at (3,4)[below right]d-k_m;
     at (3,-4)[above right]d-k_m;
     
       =q^-m/3k_0(k_0+δ)-mk_0∑_k_0≥ k_1≥⋯≥ k_m≥ 0
    q^1/2(k_0-k_m)
    q^∑_i=1^mk_i(k_i+δ)+k_i
         ×(q)_k_0+δ/(q)_k_m+δ(q)_k_0/(q)_k_0-k_1(q)_k_1-k_2⋯ (q)_k_m-1-k_m(q)_k_m [baseline=-.6ex, scale=.1] (-6,5) – (-9,5);
     (-6,-5) – (-9,-5);
     (6,5) – (9,5);
     (6,-5) – (9,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (-3,0) – (0,3) – (3,0) – (0,-3) – cycle;
     (0,3) – (0,-3);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-7,-7)[left]s;
     at (-7,7)[left]t;
     at (7,-7)[right]s;
     at (7,7)[right]t;
     at (0,6)[above]t-d+k_m;
     at (0,-6)[below]s-d+k_m;
     at (-3,4)[below left]d-k_m;
     at (-3,-4)[above left]d-k_m;
     at (3,4)[below right]d-k_m;
     at (3,-4)[above right]d-k_m;







§ APPLICATION TO THE 𝔰𝔩_3 TAIL

In this section, 
we discuss the 𝔰𝔩_3 tail of (2,2m)-torus link with one-row colorings.
A tail of a link L is a limit of the 𝔰𝔩_2 colored Jones polynomials {J_n^𝔰𝔩_2(L)}_n. 
The existence of the tail is proven for adequate links in <cit.>.
The tail of (2,2m) torus link is given by the false theta series and its explicit formula was obtained in <cit.>.
The tail or such stability of the 𝔤 colored Jones polynomials of torus knots in a case where 𝔤 is the rank 2 simple Lie algebra was shown by Garoufalidis and Vuong <cit.>. 
In the same paper, 
They gave explicit formulas of the tail for (2,2m+1)- and (4,5)-torus knots in the case of 𝔤=𝔰𝔩_3.
The author also gave two explicit formulas of the 𝔰𝔩_3 tail for the (2,2m)-torus link with anti-parallel orientation in <cit.> and it gave Andrews-Gordon type identities for a “diagonal summand” of the 𝔰𝔩_3 theta series, see also Section 9 in <cit.>.

In the following, we give an explicit formula of one-row colored 𝔰𝔩_3 Jones polynomial J_(n,0)^𝔰𝔩_3(T_⇉(2,2m)) for the (2,2m)-torus link with parallel orientation and its 𝔰𝔩_3 tail.
We use the following normalization of 𝔰𝔩_3 colored Jones polynomial for an oriented framed link.
Let D=D_1⊔ D_2⊔⋯⊔ D_p be a link diagram of a framed link L with p components through the blackboard framing. 
A coloring ω_i of D_i is given by an irreducible representations of 𝔰𝔩_3 with the highest weight (m_i,n_i) for p=1,2,…,p.
One can identify the coloring ω_i and (m_i,n_i)∈ℤ_≥ 0×ℤ_≥ 0.
Let D(ω) be the A_2 web obtained by replacing D_i with JW_ω_i where the orientation of the strand colored by m coincides with the orientation of D_i for all i=1,2,…,p. 
See Figure <ref>.












J_ω^𝔰𝔩_3(L)=D(ω)∈ℂ(q^1/6).



The above normalization of the 𝔰𝔩_3 colored Jones polynomial of a link is different from in <cit.>.



Let s and t be positive integers and d=min{s,t}.
For any integer 0≤ k≤ d,

    [baseline=-.6ex, scale=.1][rounded corners, -<-=.5] (-6,5) – (-10,5) – (-10,9) – (10,9) – (10,5) – (6,5);
    [rounded corners, -<-=.5] (-6,-5) – (-12,-5) – (-12,11) – (12,11) – (12,-5) – (6,-5);
    [->-=.5] (-6,6) – (6,6);
    [->-=.5] (-6,-6) – (6,-6);
    [->-=.5] (-6,4) to[out=east, in=north west] (0,0);
    [->-=.5] (-6,-4) to[out=east, in=south west] (0,0);
    [->-=.7] (0,0) to[out=north east, in=west] (6,4);
    [->-=.7] (0,0) to[out=south east, in=west] (6,-4);
    [fill=white] (-3,0) – (0,3) – (3,0) – (0,-3) – cycle;
     (0,3) – (0,-3);
    [fill=white] (-7,-7) rectangle (-6,-3);
    [fill=white] (-7,7) rectangle (-6,3);
    [fill=white] (7,-7) rectangle (6,-3);
    [fill=white] (7,7) rectangle (6,3);
     at (-10,7)[right]t;
     at (-12,7)[left]s;
     at (-3,4)[below left]d-k;
     at (-3,-4)[above left]d-k;
     at (3,4)[below right]d-k;
     at (3,-4)[above right]d-k;
     
    =[d-k+1]Δ(s,0)Δ(t,0)/Δ(d-k,0)

where Δ is an oriented circle.


One can calculate the above web by using the following formulas (see, for example, <cit.>).

    [baseline=-.6ex, scale=.1][->-=.3] (-12,0) – (-5,0);
    [-<-=.3] (12,0) – (5,0);
    [-<-=.7] (-5,0) to[out=north, in=west] (0,4);
    [->-=.7] (5,0) to[out=north, in=east] (0,4);
    [-<-=.7] (-5,0) to[out=south, in=west] (0,-4);
    [->-=.7] (5,0) to[out=south, in=east] (0,-4);
    [fill=white] (0,2) rectangle (1,6);
    [fill=white] (0,-2) rectangle (1,-6);
    [fill=white, xshift=-5cm] (0:2) – (120:2) – (240:2) – cycle;
    [fill=white, xshift=5cm] (60:2) – (180:2) – (300:2) – cycle;
    [fill=white] (-8,-2) rectangle (-7,2);
    [fill=white] (8,-2) rectangle (7,2);
     at (-2,3)[above]n;
     at (-2,-3)[below]n;
     at (-8,0)[above left]n;
     at (8,0)[above right]n;
        =
    [n+1]
     [baseline=-.6ex,scale=.1][->-=.8] (-5,0) – (5,0);
    [fill=white] (0,-2) rectangle (1,2);
     at (-2,0)[above]n;
     ,
     [baseline=-.6ex, scale=.1][-<-=.2] (-5,1) – (5,1);
    [-<-=.3, rounded corners] (-3,-1) – (-3,-5) – (3,-5) – (3,-1) – cycle;
    [fill=white] (0,-3) rectangle (1,3);
     at (-5,1)[above]n-k;
     at (0,-5)[below]k;
     
    =Δ(n,0)/Δ(n-k,0) [baseline=-.6ex, scale=.1][-<-=.3] (-4,0) – (4,0);
    [fill=white] (-1,-2) rectangle (0,2);
     at (4,0)[above]n-k;
        =[n+1][n+2]/[n-k+1][n-k+2] [baseline=-.6ex, scale=.1][-<-=.3] (-4,0) – (4,0);
    [fill=white] (-1,-2) rectangle (0,2);
     at (4,0)[above]n-k;




Let 
T_⇉(2,2m)=
 [baseline=-.6ex, scale=.1, yshift=-2cm][->-=.7, rounded corners] (-5,2) – (5,2) – (5,5) – (-5,5) – cycle;
[->-=.7, rounded corners] (-8,-2) – (8,-2) – (8,8) – (-8,8) – cycle;
[fill=white] (-3,-3) rectangle (3,3);
 at (0,0)2m;
be the (2,2m)-torus link with parallel orientation. 
For any one-row colorings ω_1=(s,0) and ω_2=(t,0),

    J_ω^𝔰𝔩_3(T(2,2m))=q^-m/3d(d+δ)-md
         ×∑_d≥ k_1≥⋯≥ k_m≥ 0
    q^∑_i=1^mk_i(k_i+δ)+k_iq^d-k_m(q)_d+δ/(q)_k_m+δ(q)_d/(q)_d-k_1(q)_k_1-k_2⋯ (q)_k_m-1-k_m(q)_k_m
         ×1-q^2/1-q^d-k_m+1Δ(s,0)Δ(t,0)

where d=min{s,t} and δ=| s-t |.


One can easily obtain the above formula from Theorem <ref>, Lemma <ref>, and

    Δ(n,0)=[n+1][n+2]/[2]=q^-n(1-q^n+1)(1-q^n+2)/(1-q)(1-q^2).



Let us consider a family of formal power series {f_n(q)∈ℤ[[q]]| n∈ℤ_≥ 0} and F(q)∈ℤ[[q]].
Then, we define lim_n→∞f_n(q)=F(q) if f_n(q)=F(q) in ℤ[[q]]/q^n+1ℤ[[q]] for any n.

We review an explicit formula of the 𝔰𝔩_3 tail of the (2,2m)-torus link with anti-parallel orientation
T_⇆(2,2m)=
 [baseline=-.6ex, scale=.1, yshift=-2cm][-<-=.7, rounded corners] (-5,2) – (5,2) – (5,5) – (-5,5) – cycle;
[->-=.7, rounded corners] (-8,-2) – (8,-2) – (8,8) – (-8,8) – cycle;
[fill=white] (-3,-3) rectangle (3,3);
 at (0,0)2m;.
[The 𝔰𝔩_3 colored Jones polynomial of T_⇆(2,2m) <cit.>]
For any one-row colorings ω_1=(s,0) and ω_2=(t,0),

    J_ω^𝔰𝔩_3(T_⇆(2,2m))=q^-2m/3d(d+δ)-2md
         ×∑_d≥ k_1≥⋯≥ k_m≥ 0
    q^∑_i=1^mk_i(k_i+δ)+2k_iq^2d-2k_m(q)_d+δ/(q)_k_m+δ(q)_d/(q)_d-k_1(q)_k_1-k_2⋯ (q)_k_m-1-k_m(q)_k_m
         ×(1-q)(1-q^2)/(1-q^d-k_m+1)(1-q^d-k_m+2)Δ(s,0)Δ(t,0)

where d=min{s,t} and δ=| s-t |.




[The 𝔰𝔩_3 tail of T_⇆(2,2m) <cit.>]
Let ω_n be a special coloring of T_⇆(2,2m) such that s=t=n.
Then,

    lim_n→∞f_n(q)=1/(1-q)(1-q^2)∑_k_1≥ k_2≥⋯≥ k_m≥ 0q^-2k_mq^∑_i=1^mk_i^2+2k_i/(q)_k_1-k_2(q)_k_2-k_3⋯ (q)_k_m-1-k_m(q)_k_m

where f_n(q)=q^2m/3n^2+2mnJ_ω_n^𝔰𝔩_3(T_⇆(2,2m)).



We obtain an explicit formula of the 𝔰𝔩_3 tail of T_⇉(2,2m) from Theorem <ref>
[The 𝔰𝔩_3 tail of T_⇉(2,2m)]
Let ω_n be a special coloring of T_⇉(2,2m) such that s=t=n. 
Then,

    lim_n→∞f_n(q)=1/(1-q)^2(1-q^2)∑_k_1≥ k_2≥⋯≥ k_m≥ 0q^-k_mq^∑_i=1^mk_i^2+k_i/(q)_k_1-k_2(q)_k_2-k_3⋯ (q)_k_m-1-k_m(q)_k_m

where f_n(q)=q^m/3n^2+mnq^nJ_ω_n^𝔰𝔩_3(T_⇉(2,2m)).


It is easy to see that q^k_1^2+k_1(q)_n/(q)_n-k_1=q^k_1^2+k_1∈ℤ[[q]]/q^n+1ℤ[[q]] and q^k_m^2(q)_n/(q)_k_m=q^k_m^2∈ℤ[[q]]/q^n+1ℤ[[q]].






 §.§ Acknowledgment

This work was supported by Grant-in-Aid for JSPS Fellows Grant Number 19J00252 and Grant-in-Aid for Early-Career Scientists Grant Number 19K14528.



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