Jones Polynomials and their Zeros for a Family of Knots and Links

TL;DR

This work computes the Jones polynomials $V(H_r,t)$ for an infinite family of alternating knots/links by mapping to Tutte polynomials $T(G_+(H_r),x,y)$ and evaluating at $(x,y)=(-t,-1/t)$, thereby bypassing the generic exponential complexity. The authors establish a structural expansion $V(L_{r(m)},t)=\sum_{j=1}^{N_{L,\lambda}} c_{L,j}(t)[\lambda_{L,j}(t)]^m$ and demonstrate that, for their family $H_{r(m)}$, five dominant terms contribute ($N_{H,\lambda}=5$), obtained from a generating function for $T(S_m,x,y)$. They compute explicit low-$m$ cases (e.g., $m=1,2,3$) and analyze the zeros in the complex $t$-plane, identifying a rich accumulation locus $\mathcal B$ consisting of a real line segment, a unit-circle arc, horseshoe-shaped arcs, and additional crossing arcs; endpoint positions are obtained from discriminants of an associated quartic. The results illustrate a systematic, graph-theoretic approach to Jones polynomials for infinite families and reveal intricate structures in the zeros distribution, linking knot theory with Tutte/Chromatic polynomial theory and complex analysis.

Abstract

We calculate Jones polynomials $V(H_r,t)$ for a family of alternating knots and links $H_r$ with arbitrarily many crossings $r$, by computing the Tutte polynomials $T(G_+(H_r),x,y)$ for the associated graphs $G_+(H_r)$ and evaluating these with $x=-t$ and $y=-1/t$. Our method enables us to circumvent the generic feature that the computational complexity of $V(L_r,t)$ for a knot or link $L_r$ for generic $t$ grows exponentially rapidly with $r$. We also study the accumulation set of the zeros of these polynomials in the limit of infinitely many crossings, $r \to \infty$.

Figures (1)

• Figure 1: Illustrative plot of zeros of $V(H_r,t)$ with $r=402$ in the complex $t$ plane.