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arXiv · 2507.03680

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

Yue Chen·Robert Shrock
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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, $r \to \infty$.

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Yue Chen, Robert Shrock. 2025-07-04. Jones Polynomials and their Zeros for a Family of Knots and Links. https://doi.org/10.1007/s10955-025-03531-9

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