Zeros of Jones Polynomials of Graphs

Fengming Dong, Xian'an Jin

Abstract


In this paper, we introduce the Jones polynomial of a graph $G=(V,E)$ with $k$  components as the following specialization of the Tutte polynomial:
$$J_G(t)=(-1)^{|V|-k}t^{|E|-|V|+k}T_G(-t,-t^{-1}).$$
We first study its basic properties and determine certain extreme coefficients. Then we prove that $(-\infty, 0]$ is a zero-free interval of Jones polynomials of connected bridgeless graphs while for any small $\epsilon>0$ or large $M>0$, there is a zero of the Jones polynomial of a plane graph in $(0,\epsilon)$, $(1-\epsilon,1)$, $(1,1+\epsilon)$ or $(M,+\infty)$. Let $r(G)$ be the maximum moduli of zeros of $J_G(t)$. By applying Sokal's result on zeros of Potts model partition functions and Lucas's theorem, we prove that
\begin{eqnarray*}{q_s-|V|+1\over |E|}\leq r(G)<1+6.907652\Delta_G
\end{eqnarray*}
for any connected bridgeless and loopless graph $G=(V,E)$ of maximum degree $\Delta_G$ with $q_s$ parallel classes. As a consequence of the upper bound, X.-S. Lin's conjecture holds if the positive checkerboard graph of a connected alternating link has a fixed maximum degree and a sufficiently large number of edges.


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