Quandle Coloring Quivers
Karina Cho Sam Nelson

Loop of formal diffeomorphisms and Faà di Bruno coloop bialgebra
    [
    December 30, 2023
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We consider a quiver structure on the set of quandle colorings of an oriented
knot or link diagram. This structure contains a wealth of knot and link
invariants and provides a categorification of the quandle counting invariant
in the most literal sense, i.e., giving the set of quandle colorings the
structure of a small category which is unchanged by Reidemeister moves. We
derive some new enhancements of the counting invariant from this quiver
structure and show that the enhancements are proper with explicit examples.


5.25inKeywords: Quandles, Enhancements, Quivers,
Categorified link invariants, In-degree quiver polynomials



2010 MSC: 57M27, 57M25




§ INTRODUCTION


Quandles were defined in <cit.> (see also <cit.> where
they are called distributive groupoids) as an abstract algebraic
structure generalizing the Wirtinger presentation of the knot group, with
axioms encoding the Reidemeister moves. Quandles have
been used to define many useful invariants of oriented links and knots.
In particular, associated to every finite quandle X is the positive
integer-valued quandle counting invariant Φ_X^ℤ
which can be easily computed from a diagram of an oriented knot
or link L.

Starting in <cit.> and continuing in many recent works (see
<cit.> etc.), oriented link invariants known as enhancements
of the quandle counting invariant have been defined and studied. An enhancement
is a generally stonger invariant which determines the counting invariant. The
original example is the 2-cocycle invariant where a quandle 2-cocycle
is used to define an invariant of quandle-colored oriented knots and links
known as a Boltzmann weight. The multiset of these Boltzmann weights
is then a stronger invariant which has the counting
invariant as its cardinality. For ease of comparison,
the multiset is often written as a polynomial by making the Boltzmann weights
exponents of a formal variable u with multiplicities as coefficients; in this
format we can recover the counting invariant by evaluating the polynomial
at u=1. See <cit.> for more.

Related to enhancement is the notion of categorification, whose many
recent examples include Khovanov homology in various flavors
(see e.g. <cit.>) as well as Knot Floer homology (e.g. <cit.>) and more.
The general idea is to replace basic structures with richer algebraic
structures to get finer and more sensitive information,  e.g. replacing
elements of a set with objects in a category and operations with morphisms.
In particular, categorified knot and link invariants are examples of 
enhancements of knot and link invariants.

In this paper we introduce the quandle coloring quiver, a directed
graph-valued enhancement
of the quandle counting invariant, and derive from it several
simpler enhancements. This enhancement categorifies the quandle counting
invariant in the sense that it endows the set of quandle colorings of a knot
with the structure of a small category in a way which is invariant under
Reidemeister moves.

The paper is organized as follows. In Section <ref> we recall the basics
of quandles and the quandle counting invariant. In Section <ref> we define
the quandle coloring quiver and derive from it several enhancements of the
quandle counting invariant. We provide examples which demonstrate that the new
enhancements are proper. We conclude in Section <ref> with some questions for
future work.




§ QUANDLE BASICS


We begin with some relevant definitions and examples.


A set X equipped with a binary operation  is a quandle if it satisfies

    
  * x  x = x for all x ∈ X,

  * for each y ∈ X, the map f_y: Q → Q defined by f_y(x) = x  y is a bijection, and
    
  * (x⊳ y)⊳ z = (x⊳ z)⊳(y ⊳ z) for all x,y,z ∈ X.




Let G be a group where the  operation is n-fold conjugation: x  y = y^n x y^-n. Then G is a quandle.



Let X = ℤ/nℤ where the  operation is defined by x  y ≡ 2y - x (mod n). Then X is a quandle called the dihedral quandle.


Finite quandles, which are a focus of this paper, can be fully represented by operation tables. The operation table for the dihedral quandle with n = 3 is shown below.


    [   0 1 2; 0 0 2 1; 1 2 1 0; 2 1 0 2 ]


Next, we give a combinatorial definition of the fundamental quandle of a knot or link. Let L be an oriented knot or link with n arcs in its realization. We assign a label to each arc. At each crossing, we assign the relation given by the picture below.

    0.2
        < g r a p h i c s >


Note that only the orientation of the overarc matters. The fundamental quandle Q(L) is the set of equivalence classes of quandle words generated by the arc labels under the equivalence relations given by the crossings and the quandle axioms.


We will compute Q(3_1) for the trefoil knot. Here the arcs are labelled a,b,c. 
    0.28
        < g r a p h i c s >


Then the fundamental quandle has presentation Q(3_1) = ⟨ a,b,c   |    b = c  a, a = b  c, c = a  b ⟩.
In this case, the fundamental quandle is infinite. Note that by construction, any knot or link with finitely many arcs in its diagram will have a finitely generated fundamental quandle.




Let X, Y be quandles with multiplication operations indicated by  ⊳_X and ⊳_Y respectively. A map f:X → Y is a quandle homomorphism given that f(a ⊳_X b) = f(a)⊳_Y f(b) for any a,b ∈ X.



Let L be an oriented knot or link and X a finite quandle called the coloring quandle. The coloring space Hom(Q(L),X) is the space of quandle homomorphisms from Q(L) → X. The quandle counting invariant is the cardinality of the coloring space, |Hom(Q(L),X)|, which we will denote by Φ_X^ℤ(L).



Combinatorially, each element ϕ∈Hom(Q(L),X) can be represented as a “coloring” of the diagram of L by colors from X. Using this analogy, a valid coloring is an assignment of an element from X to each arc in L's link diagram in a way that respects the quandle operation of X at each crossing. This makes sense since each arc is a generator for Q(L) and a homomorphism is uniquely defined by the mapping of generators. Since there are finitely many ways to assign colors from a finite quandle X to a link diagram with n arcs, the quandle counting invariant always yields an integer value. From here onwards we will think of homomorphisms in Hom(Q(L),X) as colorings of link diagrams.



We will compute the quandle counting invariant |Hom(Q(L),X)| where L is the trefoil as orientied in example <ref> and X is the dihedral quandle on 3 elements. Recall that Q(L) has presentation Q(L) = ⟨ a,b,c   |    b = c  a, a = b  c, c = a  b ⟩ and X has the multiplication table shown below.


    [   0 1 2; 0 0 2 1; 1 2 1 0; 2 1 0 2 ]


To count the homomorphisms, we need to count the different ways we can validly map the generators a,b,c to elements of X. As seen in example <ref>, we must satisfy c = a  b, so once colors for a and b are chosen, the color for c is determined by the multiplication table. There are 3 choices each for a and b, so we see that |Hom(Q(L),X)| = 3 · 3 = 9. These 9 colorings are depicted below where 0 is green, 1 is blue, and 2 is pink.


    0.3
        < g r a p h i c s >


For example, the coloring in the center of the grid represents the homomorphism that maps a ↦ 0, b ↦ 1, c ↦ 2.





§ QUANDLE COLORING QUIVERS


In this section we define quivers (i.e., directed graphs possibly including 
loops and bigons)
associated to a pair (L,X) consisting of an oriented link L and finite
quandle X.


Let X be a finite quandle and L an oriented link. For any
set of quandle endomorphims S⊂Hom(X,X), the associated
quandle coloring quiver, denoted 𝒬_X^S(L),
is the directed graph with a vertex
for every element f∈Hom(Q(L),X) and an edge directed
from f to g when g=ϕ f for an element ϕ∈ S. Important
special cases include the case S=Hom(X,X), which we call the
full quandle coloring quiver of L with respect to X,
denoted 𝒬_X(L),
and the case when S={ϕ}  is a singleton, which we
will denote by 𝒬_X^ϕ(L).


The vertices in 𝒬_X(L) can be identified with quandle homomorphisms
from the fundamental quandle of the link to the coloring quandle X or,
equivalently, with X-colorings of a diagram of L. Then there is a directed
edge in 𝒬_X(L) from a vertex v_1 to a vertex v_2 if there is
a quandle endomorphism e:X→ X such that the diagram of quandle
homomorphisms

    < g r a p h i c s >

commutes.


Let X be a finite quandle, S⊂Hom(X,X) and L an oriented
link. Then the  quiver 𝒬_X^S(L) is an invariant of L.



The quiver 𝒬_X^S(L) is determined up to isomorphism by X and Hom(Q(L),X) is an invariant of L.



Any invariant of directed graphs applied to 𝒬_X^S(L) defines an
invariant of oriented links.



The full quandle counting quiver forms a category with vertices as objects and
directed paths modulo loops as morphisms; hence 
𝒬_X(L) is a categorification of the quandle counting invariant
Hom(Q(L),X). The decategorification map is the forgetful functor
from the quiver to set of vertices, V(𝒬_X(L))=Hom(Q(L),X).




The Hopf link L2a1 has five colorings by the quandle X specified by
the operation table as shown below.

    [   1 2 3; 1 1 1 2; 2 2 2 1; 3 3 3 3 ]

Then for example the quandle homomorphism ϕ_1:X→ X defined by
ϕ_1(1)=2, ϕ_1(2)=1, and ϕ_1(3)=3 yields the following quandle
coloring quiver 𝒬_X^ϕ_1(L2a1)

    1
        < g r a p h i c s >
      1.25inor  0.5in
        < g r a p h i c s >

while the quandle homomorphism ϕ_2:X→ X defined by
ϕ_2(1)=ϕ_2(2)=ϕ_2(3)=3 yields
𝒬_X^ϕ_2(L2a1)

    1
        < g r a p h i c s >
      1.25inor  0.5in
        < g r a p h i c s >
    .

Setting S={ϕ_3,ϕ_4} where ϕ_3(1)=ϕ_3(2)=1 and ϕ_3(3)=2
and ϕ_4(1)=ϕ_4(2)=2 and ϕ_4(3)=1 yields the quiver
𝒬_X^S(L2a1)

    1
        < g r a p h i c s >
      1.25inor  0.5in
        < g r a p h i c s >
    .




Example <ref> inspires a simple integer-valued invariant: the number of
components of 𝒬_X^S(L). More generally, the ranks of
H_0(𝒬_X^S(L)) and H_1(𝒬_X^S(L)) are
integer-valued oriented link invariants.



Let X be the quandle given by the operation table

    [   1 2 3 4; 1 1 3 1 3; 2 4 2 4 2; 3 3 1 3 1; 4 2 4 2 4 ]
 and let ϕ:X→ X
be the endomorphism given by ϕ(1)=4, ϕ(1)=2, ϕ(1)=4
and ϕ(1)=2. Then the links L6a1 and L6a5

    < g r a p h i c s >

 both have
Φ_X^ℤ(L6a1)=Φ_X^ℤ(L6a5)=16
X-colorings and hence are not distinguished by the quandle counting
invariant defined by X; however, the quandle coloring quivers
are non-isomorphic, distinguishing the links:

    [ < g r a p h i c s > < g r a p h i c s >;         𝒬_X^ϕ(L6a1)         𝒬_X^ϕ(L6a5) ]

In particular, this example demonstrates that 𝒬_X^ϕ(L)
is a proper enhancement of the quandle counting invariant.


The full quandle coloring quiver has many multiple edges, so for simplicity
we can draw single directed edges labeled with multplicities.


The full quandle counting quiver for the link and quandle in
Example <ref> is

    < g r a p h i c s >
    .



We observe that in a quandle coloring quiver, the out-degree
deg^-(f) of every vertex is the same, namely |S|;
however, different
vertices may have different in-degrees deg^+(f). This information
can be encoded as a polynomial knot invariant:


Let X be a finite quandle, S⊂Hom(X,X) a set of quandle
endomorphisms, L an oriented link and 𝒬_X^S(L)
the associated quandle coloring quiver with set of
vertices V(𝒬_X^S(L)). Then the
in-degree quiver polynomial of L with respect to X is

    Φ_X^deg^+,S(L)
    =∑_f∈ V(𝒬_X^S(L)) u^deg^+(f).

If S={ϕ} is a singleton we will write Φ_X^deg^+,S(L)
as Φ_X^deg^+,ϕ(L) and if S=Hom(X,X) we will write
Φ_X^deg^+,S(L) as Φ_X^deg^+(L).



In Example <ref>, the Hopf link has in-degree quiver polynomial
Φ_X^deg^+(L2a1)=2u^2+u^7+2u^12 with respect to the quandle
X, and in-degree quiver polynomials
Φ_X^deg^+,ϕ_1(L2a1)=5u,
Φ_X^deg^+,ϕ_2(L2a1)=4+u^5
and
Φ_X^deg^+,S(L2a1)=2u^5+3
with respect to the quandle homomorphisms ϕ_1 and ϕ_2 and
subset S={ϕ_3,ϕ_4}.


The following example shows that in-degree quiver polynomials are proper
enhancements of the quandle counting invariant.


Let X be the quandle with operation table below. The reader can verify that
the map ϕ:X→ X defined by setting ϕ(1)=ϕ(2)=ϕ(3)=6,
ϕ(4)=4 and ϕ(5)=ϕ(6)=5 is a quandle endomorphism.

    [   1 2 3 4 5 6; 1 1 3 2 1 3 2; 2 3 2 1 2 1 3; 3 2 1 3 3 2 1; 4 4 4 4 4 4 4; 5 6 6 6 5 5 5; 6 5 5 5 6 6 6 ]

Then the links L7a3 and L7a4

    < g r a p h i c s >

 have equal quandle counting invariant
values Φ_X^ℤ(L7a3)=Φ_X^ℤ(L7a4)=30
but are distinguished by the in-degree quandle quiver polynomials

    Φ_X^deg^+,ϕ(L7a3)     =    
    22+u+2u^2+2u^3+u^4+u^6+u^9
    
     
    Φ_X^deg^+,ϕ(L7a4)     =    
    21+u+2u^2+3u^3+u^4+2u^6.




Let X be the quandle with operation table

    [   1 2 3 4 5 6 7 8; 1 1 4 2 3 3 2 1 4; 2 3 2 4 1 4 1 2 3; 3 4 1 3 2 1 4 3 2; 4 2 3 1 4 2 3 4 1; 5 8 8 8 8 5 5 5 5; 6 5 5 5 5 6 6 6 6; 7 7 7 7 7 7 7 7 7; 8 6 6 6 6 8 8 8 8 ]

and let ϕ:X→ X be given by
ϕ(1)=ϕ(2)=ϕ(3)=ϕ(4)=7, ϕ(5)=ϕ(6)=ϕ(8)=5
and ϕ(7)=6. Then we compute via our  code the
following Φ_X^deg^+,ϕ values on the prime links of up to
seven crossings from <cit.>.

    [                                                  L                                     Φ_X^deg^+,ϕ(L);                                               L2a1                         u^9 + 3u^4 + 2u^3 + u + 21;                                               L4a1                  u^16 + u^9 + 2u^4 + 2u^3 + u + 33;                                               L5a1                 2u^12 + u^9 + 3u^4 + 2u^3 + u + 43;                                               L6a1                         u^9 + 3u^4 + 2u^3 + u + 21;                                               L6a2                 2u^12 + u^9 + 3u^4 + 2u^3 + u + 43;                                               L6a3          u^16 + 2u^12 + u^9 + 2u^4 + 2u^3 + u + 55;                                               L6a4        u^27 + 3u^12 + 3u^9 + 7u^4 + 3u^3 + u + 110;                                               L6a5                 u^27 + 3u^9 + 7u^4 + 3u^3 + u + 77;                                               L6n1          u^27 + u^16 + 3u^9 + 6u^4 + 3u^3 + u + 89;                                               L7a1          u^16 + 2u^12 + u^9 + 2u^4 + 2u^3 + u + 55;                                               L7a2                  u^16 + u^9 + 2u^4 + 2u^3 + u + 33;                                               L7a3           u^16 + u^12 + u^9 + 2u^4 + 2u^3 + u + 44;                                               L7a4          u^16 + 2u^12 + u^9 + 2u^4 + 2u^3 + u + 55;                                               L7a5                         u^9 + 3u^4 + 2u^3 + u + 21;                                               L7a6                         u^9 + 3u^4 + 2u^3 + u + 21;                                               L7a7 u^27 + u^16 + 2u^12 + 3u^9 + 6u^4 + 3u^3 + u + 111;                                               L7n1                  u^16 + u^9 + 2u^4 + 2u^3 + u + 33;                                               L7n2          u^16 + u^12 + u^9 + 2u^4 + 2u^3 + u + 44. ]




The in-degree quiver polynomial is good not just for multi-component links
but can distinguish prime knots as well.


Let X be the quandle given by the operation table below.

    [   1 2 3 4 5 6 7 8 9; 1 1 3 2 7 9 8 4 6 5; 2 3 2 1 9 8 7 6 5 4; 3 2 1 3 8 7 9 5 4 6; 4 8 7 9 4 6 5 2 1 3; 5 7 9 8 6 5 4 1 3 2; 6 9 8 7 5 4 6 3 2 1; 7 6 5 4 3 2 1 7 9 8; 8 5 4 6 2 1 3 9 8 7; 9 4 6 5 1 3 2 8 7 9 ]

The prime knots 8_10 and 8_18

    < g r a p h i c s >

are then distinguished according to our  computations
by the in-degree full quiver polynomials

    Φ_X^deg^+(8_10)=9u^189 + 18u^108 + 54u^54Φ_X^deg^+(8_18)=9u^297 + 72u^54

as well as several of the in-degree quiver polynomials
associated to various endomorphisms, e.g.

    Φ_X^deg^+,ϕ(8_10)=9u^9 + 72
    Φ_X^deg^+,ϕ(8_18)=3u^27 + 78

for the quandle endomorphism defined by
ϕ(1)=ϕ(2)=ϕ(3)=9, ϕ(4)=ϕ(5)=ϕ(6)=7 and
ϕ(7)=ϕ(8)=ϕ(9)=8.






§ QUESTIONS



We end with some questions for future research.

Many more enhancements of Φ_X^ℤ can be defined from
𝒬_X; we have considered only some of the most basic. Other
examples include the eigenvalues and characteristic polynomial of the
adjacency matrix of 𝒬_X, topological invariants such as
number of components or rank of first homology, the Tutte polynomial of
the graph and much more. How are these enhancements related to other invariants?

What enhancements can be derived functorially from the categorical structure of
𝒬_X, e.g. via representations into vector space or module
categories?

If Y⊂ X is a subquandle, what is the precise relationship between 
the invariants derivable from 𝒬_Y(L) and those derivable from
 𝒬_X(L)?


abbrv



Department of Mathematics 

Harvey Mudd College

301 Platt Boulevard 

Claremont, CA 91711






Department of Mathematical Sciences 

Claremont McKenna College 

850 Columbia Ave. 

Claremont, CA 91711



