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

TL;DR

This paper computes Jones polynomials for a family of alternating knots and links using Tutte polynomials, analyzes their zeros, and addresses computational complexity issues for large crossing numbers.

Contribution

It introduces a method to efficiently compute Jones polynomials for large crossing numbers by leveraging Tutte polynomial calculations.

Findings

Successfully computed Jones polynomials for large crossing families.

Analyzed the accumulation set of polynomial zeros as crossings tend to infinity.

Circumvented exponential growth in computational complexity.

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, $r \to \infty$.