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

2025/07/04 by Yue Chen, Robert Shrock, Chen, Yue +1
Mathematics · Computer Science · #Mathematics and Applications #semigroups and automata theory #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2507.03680

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 → ∞.

Citations

• A Contribution to the Theory of Chromatic Polynomials

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• Tutte and Jones Polynomial of Link Families
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