On Yamada polynomial of spatial graphs obtained by edge replacements
Miaowang Li Fengchun Lei Fengling Li Andrei Vesnin
math.GT


On Yamada polynomial of spatial graphs]On Yamada polynomial of spatial graphs obtained by edge replacements



School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China
limiaowang@yeah.net


School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China
fclei@dlut.edu.cn


School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China
dutlfl@163.com


Novosibirsk State University and Sobolev Institute of Mathematics Novosibirsk, 630090, Russia
 vesnin@math.nsc.ru


The second author is supported in part by grants ( No.11329101 and No.11431009 ) of NSFC; the third author is supported in part by grants ( No.11671064 and No.11471151 ) of NSFC; the fourth author is supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (a grant no. 14.Y26.31.0025 of the government of the Russian Federation) and a grant RFBR-16-01-00414.

[2010]Primary 57M15; Secondary 05C31




We present formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of plane graphs, such as cycle-graphs, theta-graphs, and bouquet-graphs, by spatial parts. As a corollary, it is shown that zeros of Yamada polynomials of some series of spatial graphs are dense in a certain region in the complex plane, described by a system of inequalities. Also, the relation between Yamada polynomial of graphs and the chain polynomial of edge-labelled graphs is obtained.



[
    Andrei VESNIN
    December 30, 2023
=====================






§ INTRODUCTION


Spatial graph theory on intrinsic knotting and linking of graphs in S^3 developed in the 1980s since J.H. Conway and C. Gordon <cit.> proved that any embedding of the complete graph 𝐊_7 in ℝ^3 contains a knotted cycle and any embedding of the complete graph 𝐊_6 in ℝ^3 contains a pair of linked cycles. Being motivated by problems on knotting and linking of DNA and chemical compounds, the study of spatial graphs is in the center of interest for last decades. Thus, J. Simon <cit.> discussed the chirality of an embedding of the complete graph 𝐊_5, which answered a question raised by D.M. Walba <cit.>. C. Ernst and D.W. Sumners <cit.> introduced tangle theory of site specific recombination and gave an approach to study the behavior of DNA. X.-S. Cheng, Y. Lei and W. Yang <cit.> considered invariants of double crossover links characterize topological properties of double crossover DNA polyhedra.

The modern study of spatial graphs and their generalizations combines topological and graph-theoretical methods. The powerful of polynomial invariants of knot as well as polynomial invariants for graphs was a natural motivation for investigation of polynomial invariants of spatial graphs initiated by L.H.  Kauffman <cit.>. There are a number of invariants which connect spatial graphs and knots. Y. Ohyama and K. Taniyama <cit.> explored relations among the Vassiliev invariants of knots contained in certain graphs. In 2015, N. Chbili <cit.> gave criteria for a spatial graph to be (p,q)-lens graph. In 2017, A. Henrich and L.H. Kauffman <cit.> provided a topological invariant for pseudoknots and four-valent rigid vertex spatial graphs.

It is well known <cit.> that the Alexander ideal and Alexander polynomial are invariants of spatial graphs which are determined by the fundamental groups of the complements of spatial graphs. In 1989, S. Yamada <cit.> introduced Yamada polynomial of spatial graphs in R^3, which can distinguish some non-isotopy spatial graphs with the same fundamental group. The Yamada polynomial is an concise and useful ambient isotopy invariant for graphs with maximal degree less than four. There are many interesting results on Yamada polynomial and its generalizations. J. Murakami <cit.> investigated the two-variable extension 𝐙_S of the Yamada polynomial and gave an invariant related to the HOMFLY polynomial. In 1994, the crossing number of spatial graphs in terms of the reduced degree of Yamada polynomial has been studied by T. Motohashi, Y. Ohyama and K. Taniyama <cit.>. In 1996, A. Dobrynin and A. Vesnin <cit.> studied properties of the Yamada polynomial of spatial graphs. For any graph G, V. Vershinin and A. Vesnin <cit.> defined bigraded cohomology groups whose graded Euler characteristic is a multiple of the Yamada polynomial of G. A polynomial invariant of  virtual graphs was constructed by Y. Miyazawa <cit.> as an extension of the Yamada polynomial in 2006. Another invariant of spatial graphs associated with U_q(sl(2,C)) was introduced by S. Yoshinaga <cit.>. See <cit.> about the relation between Yamada polynomial and Yoshinaga polynomial. Nice results on the structure of the Yamada and  flows polynomials of cubic graphs are established by I. Agol and V. Krushkal <cit.>.

Zeros of polynomial invariants of knots and graphs is a question of special interest studied by A.D. Sokal <cit.> and  P. Csikvári, P.E. Frenkel, J. Hladký, T. Hubai <cit.>  for chromatic polynomial; by O.T. Dashbach, T.D. Le, X.-S. Lin <cit.> and  X. Jun, F. Zhang, F. Dong, E.G. Tay <cit.> for Jones polynomial.

This paper consists of three parts. In the first part (Sections 2 and 3) we recall properties of Yamada polynomial of graphs and obtain some formulae for computing the Yamada polynomial of graphs by edge replacements via the chain polynomial (see Theorem <ref>). In the second part (Sections 4 and 5) we give formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of  cycle graphs, theta-graphs, or bouquet graphs by spatial parts  (see Theorem <ref>). In the last part, Section 6, we prove that zeros of the Yamada polynomial of spatial graphs are dense in a certain region in the complex plane, described by a system of inequalities (see Theorem <ref>).



§ YAMADA POLYNOMIAL OF A GRAPH


We consider a graph G, admitting loops and multiple edges. Let us use standard notations p(G) and q(G) for number of vertices and number of edges of it.

Before defining the Yamada polynomial of a graph, we recall a graph invariant which is a special case of the Negami polynomial invariant <cit.>.



 <cit.>
Define 2-variable Laurent polynomial h(G) = h(G) (x,y) of graph G=(V,E), where V = V(G) is the vertex set and E = E(G) is the edge set of G, by the rule

    h(G)(x,y) = ∑_F⊂ E(-x)^-|F| f(G-F),

with f(G)=x^μ(G)y^β(G), where μ(G) and β(G) is the number of connected components of G and the first Betti number of G.




 <cit.>
The Yamada polynomial of a graph G is a Laurent polynomial in one variable, obtained by the following substitution into h(G)(x,y):

    H(G)(A) = h(G)(-1,-A-2-A^-1).





The following properties of H(x,y) hold (see <cit.> for details):


  (1) H(·)= -1.

  (2) Let e be a non-loop edge of a graph G. Then H(G)=H(G/e)+H(G-e), where G/e is the graph obtained by contracting the edge e, and G-e is the graph obtained by deleting of the edge e.

  (3) Let e be a loop edge of a graph G. Then H(G)=-σ H(G-e), where σ= A+1+A^-1.

  (4) Let G_1∪ G_2 be a disjoint union of graphs G_1and G_2, then H(G_1∪ G_2)= H(G_1)H(G_2).

  (5) Let G_1· G_2 be a union of graphs G_1 and G_2 having one common point, then
 
    H(G_1· G_2)= -H(G_1)H(G_2).


  (6) If G has an isthmus, then  H(G)= 0.


It is easy to find directly (see also <cit.>) polynomial H(G) for some simple classes of graphs.



   The following properties holds with σ = A + 1 + A^-1.


  (i) Let T_q be a tree with q edges. Then H(T_q)=0 for all q.

  (ii) Let C_n be the cycle of length n. Then H(C_n)= σ for all n.

  (iii) Let B_q be the one-vertex graph with q loops, also known as “q-bouquet”. Then H(B_q)   =  (-1)^q-1σ^q.

  (iv) Let Θ_s be the graph consisting of two vertices and s edges between them, also known as “s-theta-graph” (see Fig. <ref>). Then

    H(Θ_s)   =  1/σ+1 [σ +(-σ)^s].









Let G_1:G_2 be the union of two graphs G_1 and G_2 having only two common vertices u and v. Let K_1 and K_2 be graphs obtained from G_1 and G_2, respectively, by identifying u and v. Then

    H(G_1:G_2)     =    1/σ[ H(K_1) H(K_2) + (σ+1) H(G_1) H(G_2)             + H(K_1) H(G_2) + H(K_2) H(G_1) ].






Denote G = G_1:G_2 and prove the statement using induction by q(G), the number of edges of the graph G.

If q(G)=0,  then G_1 and G_2 consist of isolated vertices. Suppose that p(G_1) =p_1≥ 2 and p (G_2)  = p_2≥ 2. Then G consists of p_1+p_2-2 isolated vertices, that is a disjoint union of p_1+p_2-2 one-vertex graphs. By properties (1) and (4), H(G) = (-1)^p_1+p_2-2. Since p(K_1)=p_1-1 and p(K_2) = p_2-1, for the right-hand side of Eq.(<ref>) we get:

    1/σ[ (-1)^p_1 + p_2-2 + (σ + 1) (-1)^p_1 + p_2 + (-1)^p_1 -1 + p_2 + (-1)^p_2-1+p_1] = (-1)^p_1+p_2.

Hence, the equality Eq.(<ref>) holds.

Now suppose that the statement holds for all graphs which q(G) < k, where k≥ 1. Consider G=G_1:G_2 such that q(G) = k.  Let e be an edge of G. Without loss of generality, we can suppose e ∈ E(G_1). There are three cases to be considered.

Case 1. Let e be a loop of G_1. Then e is also a loop of K_1. Hence, we have

    H(G_1:G_2)      =      -σ H(G_1 - e : G_2)  
        =      - [ H(K_1 - e) H(K_2)  + (σ+1) H(G_1 - e) H(G_2) 
              + H(K_1 -e) H(G_2) + H(K_2)H(G_1-e) ]  
       =   1/σ[ H(K_1) H(K_2) + (σ+1)H(G_1) H(G_2) 
             + H(K_1) H(G_2) + H(K_2) H(G_1) ].


Case 2. Let e be neither a loop nor an edge with two end vertices u and v, where u and v are common vertices of G_1 and G_2. Then e is a non-loop edge in K_1. Hence,

    H(G/e)      =      H(G_1/e : G_2) 
         =    1/σ[  H(K_1/e )H(K_2) + (σ+1) H((G_1/e) H(G_2)  
                 + H(K_1/e) H(K_2) + H(K_2) H(G_1/e) ]

and

    H(G-e)    =     H(G_1-e:G_2) 
        =    1/σ[ H(K_1-e) H(K_2) + (σ+1) H(G_1-e) H(G_2)  
                + H(K_1-e) H(G_2) + H(K_2) H(G_1-e) ].

Therefore,

    H(G)   =     H(G/e) + H(G-e) 
       =    1/σ[ H(K_1/e) H(K_2) + (σ+1) H(G_1/e) H(G_2) 
              + H(K_1/e)H(G_2) + H(K_2)H(G_1/e) + H(K_1-e) H(K_2) 
             + (σ+1) H(G_1-e) H(G_2)+ H(K_1-e) H(G_2) 
             +H(K_2) H(G_1-e) ] .

Hence

    H(G)    =   1/σ[ ( H(K_1/e) + H(K_1-e) ) H(K_2) 
             + (σ+1) ( H((G_1/e) + H(G_1-e) ) H(G_2) 
             + ( H(K_1/e) + H(K_1-e) ) H(G_2) 
              + H(K_2)  ( H(G_1/e)+H(G_1-e) ) ]  
       =   1/σ[ H(K_1)H(K_2)+(σ+1)H(G_1)H(G_2)+H(K_1)H(G_2) 
             +H(K_2)H(G_1) ] .


Case 3. Let e be an edge with two end vertices u and v, where u and v are common vertices of G_1 and G_2. Then e is a loop in K_1. Hence,  by properties (5) and (3) we have

    H(G/e) = H(G_1/e:G_2) = -H(K_1-e)H(K_2) = 1/σH(K_1)H(K_2)

and

    H(G-e)     =    H(G_1-e:G_2) 
        =    1/σ[ H(K_1-e) H(K_2) + (σ+1) H(G_1-e) H(G_2) 
                 + H(K_1-e) H(G_2) + H(K_2) H(G_1-e) ] 
       =   1/σ[ -1/σ H(K_1) H(K_2) + (σ+1) ( H(G_1)+1/σH(K_1) ) H(G_2)  
               -1/σ H(K_1) H(G_2) + H(K_2) ( H(G_1)  + 1/σ H(K_1) ) ] .

Therefore,

    H(G)    =     H(G/e) + H(G-e)  =  1/σ H(K_1) H(K_2) 
           +  1/σ[ - 1/σ H(K_1) H(K_2)  + (σ+1) H(G_2) ( H(G_1) + 1/σ H(K_1) ) 
                -1/σ H(K_1) H(G_2) + H(K_2) ( H(G_1) + 1/σ H(K_1) ) ] 
       =   1/σ[ H(K_1) H(K_2) + (σ+1) H(G_1) H(G_2) + H(K_1) H(G_2)
             + H(K_2) H(G_1) ] .

This completes the proof.




§ YAMADA POLYNOMIALS OF A GRAPH OBTAINED BY EDGE REPLACEMENTS


We recall some properties of the chain polynomial introduced by R.C. Read and E.G. Whitehead <cit.>, see also <cit.>.  The chain polynomial is  defined on edge-labelled graphs with labels are elements of a commutative ring with unity. We will denote the edges by the labels associated with them.


The chain polynomial Ch(G) of a labelled graph G is defined as

    Ch(G) = ∑_Y ⊂ E F_G-Y (1-w) ∏_a ∈ Y a,

where the sum is taken over all subsets of the edge set E of G, F_G-Y (1-w) denotes the flow polynomial of the subgraph G - Y calculated at 1-w, and ∏_a∈ Y denotes the product of edge-labels of Y.


The chain polynomial can be also defined in the following recursive form.

 
The chain polynomial  Ch (G) (w) in a variable w of a labelled graph G is defined by following rules.


  (1) If G is edgeless, then Ch(G)=1.

  (2) Otherwise, suppose a is an edge of G labelled by a. Then


  (2a) If a is a loop, then Ch(G) = (a-w) Ch(G-a).

  (2b) If a is not a loop, then Ch(G) = (a-1) Ch (G-a) + Ch(G/a).





Let C_n be the n-cycle with edges labelled by a_1, a_2, ⋯, a_n, then

    Ch(C_n) = ∏_i=1^n a_i   -   w.




Let Θ_s be the s-theta-graph with edges labelled by a_1, a_2, ⋯, a_s, then

    Ch (Θ_s) = 1/1-w[ ∏^s_i=1(a_i-w) - w ∏ ^s_i=1(a_i-1) ].




Let B_q be the q-bouquet with q loops labelled by a_1, a_2, …,  a_q, then

    Ch ( B_q) = ∏^_i=1(a_i-w).



To explore the relation between the chain polynomial and the Yamada polynomial, inspired by <cit.>, let us give the following notation. Let G be a connected labelled graph. Denote by G the graph obtained from G by replacing each edge a= uv of G by a connected graph K_a with two attached vertices u and v that has only the vertices u and v in common with G-a.

Let K'_a be the graph obtained from K_a by identifying u and v, the two attached
vertices of K_a. Denote

    α_a = α(K_a) := 1/σ[(σ+1)H(K_a)+H(K'_a)],


    β_a = β(K_a) := 1/σ[H(K_a)+H(K'_a)],

and

    γ_a = γ(K_a) : =1-α (K_a)/β (K_a).

It is easy to see that

    H(K'_a)=(σ+1)β_a-α_a    and      H(K_a)=α_a-β_a.


The following result gives the relation between Yamada polynomial and chain polynomial.


Let G be a connected labelled graph, and G be the graph obtained from G by replacing the edge a by a connected graph K_a for every edge a of G. If we replace w by -σ, and replace a by γ_a for every label a in Ch(G), then we get

    H(G)=∏_a∈ E(G)β_a/(-1)^q(G)-p(G)Ch(G),

where p(G) and q(G) are the numbers of vertices and edges of G, respectively.



Denote

    T(G) = (-1)^q(G)-p(G) H(G)/∏_a∈ E(G)β_a.

To prove the statement we will show that T(G) satisfies conditions from Definition 3.2 of Ch(G) if we identify w with -σ and a with γ_a. Let us check these conditions.

(1) If G is edgeless, then E(G) = ∅, E(G) = ∅ and  H(G) =  (-1)^p(G) = (-1)^p(G). Then

    T(G)=(-1)^q(G)-p(G) H(G)  /∏_a∈ E(G)β_a= (-1)^-p(G) (-1)^p(G) = 1.


(2a) If a is a loop of G, then

    T(G)     =    (-1)^q(G)-p(G) H(G)/∏_a∈ E(G)β_a
        =    (-1)^q(G-a)+1-p(G-a)[-H(K_a') H(G-a)]/β_a∏_a∈ E(G-a)β_a
        =    (-1)^q(G-a)+1-p(G-a)[ ( α_a - (σ+1) β_a) H(G-a) ]/β_a∏_a∈ E(G-a)β_a
       =   - ( α_a-(σ+1)β_a) )/β_aT(G-a) =(γ_a+σ)T(G-a).

Compare with Ch(G) = (a - w) Ch (G-a) from Definition 3.2.

(2b) If a ∈ E(G) is not a loop, then by Proposition <ref>

    H(G)    =    H(G-a:K_a) 
        =    1/σ[ H(G/a) H(K'_a )+ (σ+1) H(G-a) H(K_a) 
                          + H(K'_a) H(G-a) + H(K_a) H(G/a)] 
       =    1/σ[ H(G/a) ( H(K'_a)+H(K_a) ) 
                        + H(G-a)( (σ+1)H(K_a)+H(K'_a)  ) ] 
       =   β_aH(G/a) + α_aH(G-a).

Therefore,

    T(G)     =     (-1)^q(G)-p(G)H(G)/∏_a∈ E(G)β_a
        =    (-1)^q(G-a)+1-p(G-a)[ β_a H(G/a) + α_a H(G-a) ]/β_a∏_a∈ E(G-a)β_a
       =   (-1)^q(G/a)+1-p(G/a)-1β_a H(G/a) /β_a∏_a∈ E(G/a)β_a
             +
    (-1)^q(G-a)+1-p(G-a)α_aH(G-a) /β_a∏_a∈ E(G-a)β_a
       =   T(G/a) + (-α_a/β_a) T(G-a) .

Thus,

    T(G) = T(G/a) +(γ_a-1) T(G-a).

Compare with Ch (G) = Ch (G/a) +  (a-1) Ch (G-a) from Definition 3.2.




§ YAMADA POLYNOMIAL OF SPATIAL GRAPHS


Next we will consider the Yamada polynomial of spatial graphs. Through the paper we work in the piecewise-linear category. Let G be a graph embedded in ℝ^3, and g be a diagram of  G. For any double point, S. Yamada <cit.> defined the spin of +1, -1 and 0, which are denoted by S_+, S_- and S_0, as shown in Fig. <ref>.



Let S be the plane graph obtained from g by replacing each double point with a spin.  S is called a state on g. Denote the set of all states by U(g). Put

    c(g|S)= A^m_1-m_2,

where m_1 and m_2 are the numbers of double points with spin S_+ and S_-, respectively, used to obtain S from g.



 <cit.>
The Yamada polynomial of a diagram g of a spatial graph G is a Laurent polynomial in variable A defined as follows

    R[g] = R[g](A) : =∑_S∈ U(g) c(g|S)H(S),

where H(S) = h(S)(-1,-A-2-A^-1).


For interesting properties of the Yamada polynomial and its invariantness under transformations of spatial graph diagrams see <cit.>. Here we recall only some of them.



 <cit.>
The following properties hold.


  (1) Let g_1∪ g_2 be a disjoint union of diagrams g_1 and g_2, then

    R[g_1∪ g_2]= R[g_1]R[g_2].


  (2)  Let g_1· g_2 be a union of diagrams g_1 and g_2 having one common point, then

    R[g_1· g_2] = -R[g_1]R[g_2].


  (3) If g has an isthmus, then  R[g]= 0.




If a diagram g of G does not have double points, then R[g]= H(G).


Proposition <ref> about H(G) implies the similar  formula for polynomial R[g].


Let  g_1:g_2 be the union of two diagrams g_1 and g_2 having only two common vertices u and v. Let k_1 and k_2 be diagrams obtained from g_1 and g_2, respectively, by identifying u and v. Then

    R[g_1:g_2] = 1/σ( R[k_1] R[k_2] + (σ+1) R[g_1] R[g_2] 
     + R[k_1] R[g_2] + R[k_2] R[g_1] ).




By Definition <ref> and Proposition <ref>, we obtain

    R[g_1:g_2]     =    ∑_S∈ U(g_1:g_2) c(g_1:g_2|S) H(S) 
       =   ∑_S_1∈ U(g_1)∑_S_2∈ U(g_2) c(g_1|S_1) c(g_2|S_2) H(S_1:S_2) 
       =   ∑_S_1∈ U(g_1)∑_S_2∈ U(g_2) c(g_1|S_1) c(g_2|S_2) 1/σ[ H(S_1')H(S_2') 
             + (σ+1) H(S_1) H(S_2) +H(S_1)H(S_2')+H(S_1')H(S_2) ]
       =   1/σ(  ∑_S_1∈ U(g_1)[ c(g_1|S_1) H(S_1') ] ∑_S_2∈ U(g_2)[ c(g_2|S_2) H(S_2') ]  
            + (σ+1) ∑_S_1∈ U(g_1)[ c(g_1|S_1) H(S_1) ] ∑_S_2∈ U(g_2)[ c(g_2|S_2) H(S_2) ] 
            +  ∑_S_1∈ U(g_1)[ c(g_1|S_1)H(S_1') ]  ∑_S_2∈ U(g_2)[ c(g_2|S_2) H(S_2) ] 
           +∑_S_1∈ U(g_1)[ c(g_1|S_1) H(S_1) ]  ∑_S_2∈ U(g_2)[c(g_2|S_2) H(S_2') ] ) 
       =   1/σ[ R[k_1]R[k_2] + (σ+1) R[g_1] R[g_2] + R[k_1] R[g_2] 
                   + R[k_2] R[g_1] ].

This completes the proof.





§ YAMADA POLYNOMIALS OF DIAGRAMS OBTAINED BY EDGE REPLACEMENTS


Let G be a connected plane labelled graph, where each edge a with terminal vertices u and v is labelled by a diagram g_a having  two vertices u_a* and v_a* indicated. We define G(g_a) to be the spatial graph (its diagram) obtained from G by replacing an edge a= u_av_a of G by a connected diagram g_a with identification u_a with u_a* and v_a with v_a* in such a way that  g_a has only vertices u_a=u_a* and v_a=v_a* in common with G-a. We define G(g_a, g_b) to be the spatial graph (its diagram) obtained from G(g_a) by replacing an edge b= u_bv_b of G(g_a)∩ (G-a)  by a connected diagram g_b with identification u_b with u_b* and v_b with v_b* in such a way that  g_b has only vertices u_b=u_b* and v_b=v_b* in common with G(g_a). With the same construction, we can obtain G(g_a_1, g_a_2, g_a_3, …).
In this context we denote by g_a' the diagram obtained from g_a by identifying vertices u_a* and v_a* in such a way that no new intersections appear.

 The following properties hold.


  (1) Let C_n be the n-cycle graph with edges labelled by a_1, a_2, …, a_n, then

    R[C_n(g_a_1, g_a_2, ⋯, g_a_n)] = ∏_i=1^n(-R[g_a_i])+σ∏_i=1^n ( R[g_a_i]+R[g_a_i']/σ).


  (2) Let Θ_s be the s-theta-graph with edges labelled by a_1, a_2, …, a_s, then

    R[Θ_s(g_a_1, g_a_2, ⋯, g_a_s)]                                   
    
    = (-1)^s/1+σ∏_i=1^sR[g_a_i'] + σ/1+σ∏_i=1^s (σ+1/σR[g_a_i]+1/σR[g_a_i'] ).


  (3) Let B_q be the q-bouquet with loops labelled by a_1, a_2, …, a_q, then

    R[B_q(g_a_1, g_a_2, ⋯, g_a_q)] = (-1)^q-1∏_i=1^q R[g_a_i'].





(1) Firstly, we prove Eq.(<ref>) by induction on n. Recall, that by Example <ref>, the chain polynomial of the n-cycle graph is 
    Ch (C_n) = ∏_i=1^n a_i-w.


If n=1, then R[C_1(g_a_1)]=R[g_a_1'], so Eq.(<ref>) holds.

Suppose  Eq.(<ref>) holds for all n ≤ k-1, where k ≥ 2. Observe that spatial graph  C_k(g_a_1, g_a_2, …, g_a_k) can be presented as the union g_a_k : L_k-1(g_a_1, g_a_2, …, g_a_k-1), where L_k-1 is a graph with k vertices connected by k-1 edges one by one, as shown in Fig. <ref>.



Then

    R[C_k(g_a_1, g_a_2, ⋯, g_a_k)]=R[g_a_k : L_k-1(g_a_1, g_a_2, ⋯, g_a_k-1)].

By Proposition <ref>, we have

    R[g_a_k : L_k-1(g_a_1, g_a_2, ⋯, g_a_k-1)]
                       = 1/σ[ R[L_k-1(g_a_1, g_a_2, ⋯, g_a_k-1)']R[g_a_k'] 
                           + (1+σ)R[L_k-1(g_a_1, g_a_2, ⋯, g_a_k-1)]R[g_a_k]
                             + R[L_k-1(g_a_1, g_a_2, ⋯, g_a_k-1)']R[g_a_k] 
                           +R[L_k-1(g_a_1, g_a_2, ⋯, g_a_k-1)]R[g_a_k'] ].


Applying Proposition <ref>, we get

    R[L_k-1(g_a_1, g_a_2, ⋯, g_a_k-1)] = (-1)^k∏_i=1^k-1R[g_a_i]

and

    R[L_k-1(g_a_1, g_a_2, ⋯, g_a_k-1)'] = R[C_k(g_a_1, g_a_2, ⋯, g_a_k-1)] 
     =∏_i=1^k-1( -R[g_a_i] ) + σ∏_i=1^k-1( R[g_a_i] + R[g_a_i']/σ).

Therefore,

    R[C_k(g_a_1, g_a_2, ⋯, g_a_k)]
           = 1/σ[  ( ∏_i=1^k-1(-R[g_a_i]) + σ∏_i=1^k-1R[g_a_i]+R[g_a_i']/σ) R[g_a_k'] 
                   +(1+σ)(-1)^k( ∏_i=1^k-1 R[g_a_i] ) R[g_a_k] 
                    + ( ∏_i=1^k-1(-R[g_a_i]) + σ∏_i=1^k-1R[g_a_i] + R[g_a_i']/σ) R[g_a_k] 
                   +(-1)^k( ∏_i=1^k-1R[g_a_i] ) R[g_a_k'] ]  
           = ∏_i=1^k (-R[g_a_i]) + σ∏_i=1^k  ( R[g_a_i] +R[g_a_i']/σ).


Thus, Eq.(<ref>) holds.

(3) Similarly to the above case, Eq.(<ref>) is equivalent to

    R[B_q(g_a_1, g_a_2, …, g_a_q)]                                 
    
    =R[g_a_1]   R[g_a_2] ⋯ R[g_a_q-1]   R[g_a_q]= (-1)^q-1∏_i=1^qR[g_a_i'].


(2) Now, we prove Eq.(<ref>) by induction on s.  If s=1, then R[Θ_1(g_a_1)]= R[g_a_1] and Eq.(<ref>) holds.

Suppose that Eq.(<ref>) holds for all s ≤ k-1, where k≥ 2. Observe that the left-hand side of Eq.(<ref>) can be presented as the union R[Θ_s(g_a_1, g_a_2, ⋯, g_a_s-1):  g_a_s].

Since

    R[Θ_s(g_a_1, g_a_2, …, g_a_s)'] = R[B_s(g_a_1, g_a_2, …, g_a_s)],

by Proposition <ref> we have

    R[Θ_s(g_a_1, g_a_2, …, g_a_s)]                             
    
     = 1/σ( R[Θ_s(g_a_1, g_a_2, …, g_a_s-1)']R[g_a_s'] 
         + (1+σ) R[Θ_s(g_a_1, g_a_2, …, g_a_s-1)]R[g_a_s] 
         +R[Θ_s(g_a_1, g_a_2, …, g_a_s-1)'] R[g_a_s]
     + R[Θ_s(g_a_1, g_a_2, …, g_a_s-1)]R[g_a_s'] ) 
    
     = (-1)^s/1+σ∏_i=1^sR[g_a_i'] + σ/1+σ∏_i=1^s ( σ+1/σR[g_a_i] + 1/σR[g_a_i'] ).

The proof is competed.


If every edge a of G is labelled by the same diagram g_a = g, we denote G(g, …, g) shortly by G(g).


If all edges of a graph G are labelled by the same diagram g, then

    R[C_n(g)]=(-R[g])^n+σ( R[g]+R[g']/σ)^n,


    R[Θ_s(g)] = (-1)^s/1+σR[g']^s + σ/1+σ( σ+1/σR[g]+1/σR[g'] )^s,

and

    R[B_q(g)] = (-1)^q-1 R[g']^q.




Remark that  C_n(g_a_1, g_a_2, …, g_a_n) is the “ring of beads” discussed in <cit.> and <cit.>. By Eq.(<ref>), we get the following property.


The Yamada polynomial R[C_n(g_a_1, g_a_2, …, g_a_n)] of a  “ring of beads”  graph C_n(g_a_1, g_a_2, …, g_a_n) is
independent of the order in which the beads occur.


The following example illustrates the applying of Theorem <ref>.


Let ∞_+ be the spatial graph diagram with two vertices and one double point signed by “+”, see Fig. 2. We denote the mirror image of diagram ∞_+ by ∞_-. Direct calculations give Yamada polynomials R[∞_+]=A^-2σ and R[∞_+']=σ, where σ = A + 1 + A^-1. By Theorem <ref>, the Yamada polynomial of C_n(∞_+) is as follows:

    R[C_n(∞_+)]=(-A^-2σ)^n+σ(A^-2+1)^n;

the Yamada polynomial of Θ_s(∞_+) is as follows:

    R[Θ_n(∞_+)]=(-σ)^s+σ((σ+1)A^-2+1)^s/1+σ;

and the Yamada polynomial of B_q(∞_+) is as follows:

    R[B_q(∞_+)]=(-1)^q-1σ^q.







§ ZEROS OF YAMADA POLYNOMIAL


In this section, we investigate the density of zeros of Yamada polynomial of two classes of spatial graphs. One is the “ring of beads” graphs C_n(Θ_s) obtained from C_n by replacing the edge a by a connected plane s-theta graph Θ_s for every edge a of C_n. The other is the class of spatial graphs C_n(Θ_s(∞_+)) and their  mirror images C_n(Θ_s(∞_-)).

To study zeros of families of polynomials we will base on the following  results by S. Beraha, J. Kahane, N.J. Weiss <cit.> and by A.D. Sokal <cit.>.

 <cit.>
If {f_n(x)} is a family of polynomials such that

    f_n(x) = α_1(x) λ_1(x)^n + α_2(x) λ_2(x)^n +… + α_l(x) λ_l(x)^n,

where the α_i(x) and λ_i(x) are fixed non-zero polynomials, such that no pair i≠ j has
λ_i(x) ≡ωλ_j(x) for some complex number ω of unit modulus. Then z is a limit of zeros of  {f_n(x)} if and only if


  (1) two or more of the λ_i(z) are of equal modulus, and strictly greater in modulus than the others; or

  (2) for some j, the modulus of λ_j(z) is strictly greater than those of the others, and α_i(z)=0.



 <cit.> 
Let F_1, F_2, and G be analytic functions on a disc { z ∈ℂ : |z|<R } such that |G(0)|≤ 1 and G not constant. Then, for each ϵ >0 there exists s_0<∞ such that for all integers s≥ s_0 the equation

    |1+F_1(z)G(z)^s|=|1+F_2(z)G(z)^s|

has a solution in the disc |z|<ϵ.


Now, let us turn to families of Yamada polynomials.



Zeros of the Yamada polynomials {R[C_n(Θ_s)]}_n=1,s=1^∞, ∞ are dense in the region {z∈ℂ  :    |z+1+z^-1|≥ 1}.



By calculating the Yamada polynomial of C_n(Θ_s)  we obtain

    R[C_n(Θ_s)]=H(C_n(Θ_s)) = ( -H(Θ_s) )^n + σ( H(Θ_s)+H(Θ_s')/σ)^n.

We will apply Theorem <ref> in the case l=2 and use condition (1). Thus, numbers A satisfying the equation

    |-H(Θ_s)| = | H(Θ_s) + H(Θ_s')/σ|,

are exactly limits of zeros of Yamada polynomials  { R[C_n (Θ_s)] }_n=1,s=1^∞, ∞.

Observe that Θ_s' is the s-bouquet B_s. Since polynomials H(Θ_s) and H(B_s) were calculated in Lemma <ref>, we have

    H(Θ_s)=σ-(-1)^s-1σ^s/1+σ    and     H(Θ_s')=(-1)^s-1σ^s.

Therefore, Eq.(<ref>) is equivalent to

    | σ-(-1)^s-1σ^s/1+σ| =  | 1/σ[ σ-(-1)^s-1σ^s/1+σ+(-1)^s-1σ^s ] | ,

whence

    |σ-(-1)^s-1σ^s| =   |1+(-1)^s-1σ^s|

and then

    |1+σ(-σ)^-s|=|1-(-σ)^-s|.


Recall that σ = A + 1 + A^-1. Let A_0 be a complex number with |A_0+1+A_0^-1|≥ 1. Denoting a=A-A_0 we get that  Eq.(<ref>) is equivalent to

    | 1+ ( (a+A_0) + (a+A_0)^-1 +1 )  (-(a+A_0) - (a +A_0)^-1 - 1 )^-s| 
    
    = | 1- ( - (a +A_0)-(a +A_0)^-1 -1)^-s|.


Consider function

    G(a) = (-(a+A_0) - (a+  A_0)^-1 -1 )^-1.

Then Eq.(<ref>) is equivalent to

    | 1+ ( (a+A_0) + (a+A_0)^-1 +1 )  G(a)^s| 
    
    = | 1- G(a)^s|

and |G(0)|=1/|A_0+1+A_0^-1|≤ 1. Hence by Lemma <ref>, for any positive real  number ε > 0 there exists s_0 such that for any s≥ s_0 the equation Eq.(<ref>) has a solution in the disk |a| < ε. Thus, for A_0 there exists  A' such that  |A'-A_0|<ε/2 and A'  satisfies the equation Eq.(<ref>).

Since zeros of Eq.(<ref>) are the limits of zeros of Yamada polynomials  { R [C_n(Θ_s)] }_n=1,s=1^∞, ∞,  there exist integers n>0 and s>0 such that R[C_n(Θ_s)] has zero A^* with |A^*-A'|< ε/2.

So there exists a zero A^* of R[C_n(Θ_s)] with |A^*-A_0|≤ε, where complex number A_0 satisfies |A_0+1+A_0^-1|≥ 1.


We remark that another proof of Theorem <ref> follows from <cit.> by the relations between graph polynomials. Indeed, Therem 7.2 of <cit.> states that if |q_0 -1| ≥ 1, then for each ε >0 there exists s_0 < ∞ and n_0(s) < ∞ such that for all s ≥ s_0 and n ≥ n_0, the flow polynomial F_C_n(Θ_s) (q) has a zero in the disc | q - q_0| < ε. Recall, see for example <cit.>, that for planar graphs G the Yamada polynomial coincides with a renormalization of the flow polynomial:

    F_G(q) = (-1)^V(G) - E(G) R_G (A),     where   q = A +2 + A^-1

with |q_0 - 1| ≥ 1 equivalent to |A_0 + 1 + A^-1_0| ≥ 1.

Next we consider zeros of Yamada polynomial of the second class of spatial graphs.


Zeros of Yamada polynomials

    {R[C_n(Θ_s(∞_+))]}_n=1, s=1^∞, ∞

are dense in the region

    {z∈ℂ  :    |z^-3+2z^-2+z^-1+1|≤ |z^-1+1+z|}.

Zeros of the Yamada polynomials { R[C_n(Θ_s(∞_-))] }_n=1, s=1^∞, ∞ are dense in the region

    {z∈ℂ  :    |z^3+2z^2+z^1+1|≤ |z^-1+1+z|}.





By calculating the Yamada polynomial of C_n(Θ_s(∞_+))  we obtain

    R[C_n(Θ_s(∞_+))]= ( -R[Θ_s(∞_+)] )^n + σ( R[Θ_s(∞_+)]+R[Θ_s(∞_+)']/σ)^n.

By Theorem <ref>, numbers A satisfying the equation

    |-R[Θ_s(∞_+)]| = | R[Θ_s(∞_+)] + R[Θ_s(∞_+)']/σ|,

are exactly limits of zeros of Yamada polynomials  { R[C_n (Θ_s(∞_+))] }_n=1,s=1^∞, ∞.

Observe that Θ_s' is the s-bouquet B_s. By Example <ref> we have

    R[Θ_s(∞_+)]=(-σ)^s+σ((σ+1)A^-2+1)^s/1+σ

and

    R[Θ_s(∞_+)']=(-1)^s-1σ^s.

Therefore, Eq.(<ref>) is equivalent to

    | (-σ)^s+σ((σ+1)A^-2+1)^s/1+σ|                                 
    
    =  | 1/σ( (-σ)^s+σ((σ+1)A^-2+1)^s/1+σ+(-1)^s-1σ^s ) | ,

whence

    |1+σ( (σ+1)A^-2+1/-σ)^s|=|1-( (σ+1)A^-2+1/-σ)^s|.


Recall that σ = A + 1 + A^-1. Let A_0 be a complex number with

    |A_0^-3+2A_0^-2+A_0^-1+1|≤ |A_0^-1+1+A_0|.

Denoting a=A-A_0 we get that  Eq.(<ref>) is equivalent to

    | 1+ ( (a+A_0)+1+(a+A_0)^-1) .                 
    ·. ( ( (a+A_0)+2+(a+A_0)^-1)(a+A_0)^-2+1/-( (a+A_0)+1+(a+A_0)^-1))^s | 
    
    = | 1- ( ( (a+A_0)+2+(a+A_0)^-1)(a+A_0)^-2+1/-( (a+A_0)+1+(a+A_0)^-1))^s |.


Consider the function

    G(a) =  ( (a+A_0)+2+(a+A_0)^-1)(a+A_0)^-2+1/-( (a+A_0)+1+(a+A_0)^-1) .

Then Eq.(<ref>) is equivalent to

    | 1+ ( (a+A_0) + (a+A_0)^-1 +1 )  G(a)^s| 
    
    = | 1- G(a)^s|

and |G(0)|=| (A_0+2+A_0^-1)A_0^-2+1/A_0+1+A_0^-1| ≤ 1. Hence by Lemma <ref>, for any positive real  number ε > 0 there exists s_0 such that for any s≥ s_0 the equation Eq.(<ref>) has a solution in the disk |a| < ε. Thus, for A_0 there exists  A' such that  |A'-A_0|<ε/2 and A'  satisfies the equation Eq.(<ref>).

Since zeros of Eq.(<ref>) are the limits of zeros of Yamada polynomials  { R [C_n(Θ_s(∞_+))] }_n=1,s=1^∞, ∞,  there exist integers n>0 and s>0 such that R[C_n(Θ_s(∞_+))] has zero A^* with |A^*-A'|< ε/2.

So there exists a zero A^* of R[C_n(Θ_s(∞_+))] with |A^*-A_0|≤ε, where complex number A_0 satisfies |A_0^-3+2A_0^-2+A_0^-1+1|≤ |A_0^-1+1+A_0|.

By Proposition 6 in <cit.>, the relation between the Yamada polynomials of a graph g and its mirror image of a diagram g  is R[g](A)=R[g](A^-1).

Since C_n(Θ_s(∞_-)) is the mirror image of diagram C_n(Θ_s(∞_+)), the zeros of R[C_n(Θ_s(∞_-))] are dense in the following region {z∈ℂ  :    |z^3+2z^2+z^1+1|≤ |z^-1+1+z|}.



Combining Theorem <ref> and Theorem <ref>, we have


Zeros of the Yamada polynomial of spatial graphs are dense in the following region:

    Ω =
    { z ∈ℂ  :   |z+1+z^-1| ≥min{1, |z^3+2z^2+z+1|, |1 + z^-1 + 2 z^-2 + z^-3 | }}.



 
  
 
 
  
   
   
   
  






§ ACKNOWLEDGMENTS


The authors are thankful to Prof. Andrey Dobrynin for the helpful Maple's visualization of regions, discussed in the paper.

amsplain
10

AK
I. Agol, V. Krushkal, Structure of the flow an Yamada polynomial of cubic graphs, preprint available at arXiv:1801.00502v1.

BJ
S. Beraha, J. Kahane, N.J. Weiss, Limits of zeros of recursively defined families of polynomials. Studies in Foundations and Combinatorics,  Advances in Mathematics Supplementary Studies 1 (1978), 212–232.

CC
J.H. Conway, C. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983), 445–453.

NC
N. Chbili, The Yamada polynomial of lens spatial graphs,  Asian-European Journal of Mathematics 8.02 (2015), 1550029.

CLY
X.-S. Cheng, Y. Lei, W. Yang, The Homfly polynomial of double crossover links, J. Math. Chem. 52 (2014), 23–41.

CFHH
P. Csikvári, P. Frenkel, J. Hlafký, T. Hubai, Chromatic roots and limit of dense graphs, Discrete Mathematics 340 (2017), 1129–1135.

DLL
O.T. Dashbach, T.D. Le, X.-S. Lin, Quantum Morphism and the Jones Polynomial, Commun. Math. Phys. 224 (2001), 427–442.

DV
A. Dobrynin, A. Vesnin, On Yamada polynomial for graphs embedded knot-wise in the three-dimensional space, Vychisl. Sistemy 155 (1996), 37–86. (in Russian). The translation is available at  https://www.researchgate.net /publication/266336562.

Kobe
A. Dobrynin, A. Vesnin, On the Yoshinaga polynomial of spatial graphs, Kobe J. Math. 20:1-2 (2003), 31–37.

ES
C. Ernst, D.W. Sumners, A calculus for rational tangles: applications to DNA recombination, Mathematical Proceedings of the Cambridge Philosophical Society 108(3) (1990), 489–515.

HK
A. Henrich, L.H. Kauffman, Tangle insertion invariants for pseudoknots, singular knots, and rigid vertex spatial graphs. In Knots, Links, Spatial Graphs, and Algebraic Invariants, Contemporary Mathematics 689 (2017), 177–189.

xj
X. Jin, F. Zhang, On computing Kauffman bracket polynomial of Montesinos links, Journal of Knot Theory and Its Ramifications 19(08) (2010), 1001–1023.

XF
X. Jin, F. Zhang, F. Dong, E.G.Tay,  Zeros of the Jones Polynomial are Dense in the Complex Plane, Electronic Journal of Combinatorics 17(1) (2010), 2493–2503.

K
L.H. Kauffman, A Tutte polynomial for signed graphs, Discrete Applied Mathematics 25(1) (1989), 105–127.

k
L.H. Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311(2) (1989), 697–710.

K1
S. Kinoshita, Alexander polynomials as isotopy invariants. I, Osaka Math. J. 10(2) (1958), 263–271.

K2
S. Kinoshita, Alexander polynomials as isotopy invariants. II, Osaka Math. J.  11 (1959), 91–94.

M
J. Murakami, The Yamada polynomial of spacial graphs and knit algebras,  Communications in Mathematical Physics 155(3) (1993), 511–522.

MY
Y. Miyazawa, The Yamada polynomial for virtual graphs, in Intelligence of Low Dimensional Topology, 2006 – The International Conference Hiroshima, Japan, 22–26 July 2006. Eds. J.S. Carter, S. Kamada, L.H. Kauffman, A. Kawauchi, T. Kohno.  (2007), 205–212.

MO
T. Motohashi, Y. Ohyama, K. Taniyama, Yamada polynomial and crossing number of spatial graphs, Rev. Mat. Univ. Complut. Madrid 7(2) (1994), 247–277.

N
S. Negami, Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987), 601–622.

OT
Y. Ohyama, K. Taniyama, Vassiliev invariants of knots in a spatial graph, Pacific J. Math. 200(1)  (2001), 191–205.

RE
R.C. Read, E.G. Whitehead, Chromatic polynomials of homeomorphism classes of graphs, Discrete. Math. 204 (1999), 337–356.

SO
A.D. Sokal, Chromatic roots are dense in the whole complex plane, Comb. Probab. Comput. 13 (2004), 221–261.

S
J. Simon, Topological chirality of certain molecules, Topology 25(2) (1986), 229–235.

VV
V. Vershinin, A. Vesnin, Yamada Polynomial and Khovanov Cohomology, in Intelligence of Low Dimensional Topology, 2006 – The International Conference Hiroshima, Japan, 22–26 July 2006. Eds. J.S. Carter, S. Kamada, L.H. Kauffman, A. Kawauchi, T. Kohno.  (2007), 337–346.

W
D. M. Walba, Stereochemical topology, in Chemical applications of topology and graph theory: A Collection of Papers from a Symposium Held at University of Georgia, Athens, Georgia, USA, 18-22 April 1983, ed. R.B. King  (Elsevier, 1983), 17–32.

YA
S. Yamada, An invariant of spatial graphs, J. Graph Theory 13 (1989), 537–551.

YO
S. Yoshinaga, An invariant of spatial graphs associated with U(q)(sl(2, C), Kobe J. Math. 8 (1991), 25–40.




