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October 20, 2025Open Access

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

Key Points

Jones polynomials are calculated for knots and links by evaluating Tutte polynomials for associated graphs.
Evaluating Jones polynomials using the method provided significantly reduces computational complexity tied to crossings.
The study explores the accumulation set of zeros as crossings increase infinitely, enhancing understanding of polynomial behavior.
Understanding zeros of Jones polynomials in this context may reveal deeper relationships within knot theory and graph structures.

Abstract

We calculate Jones polynomials V (Hᵣ, t) for a family of alternating knots and links Hᵣ with arbitrarily many crossings r, by computing the Tutte polynomials T (G_+ (Hᵣ), x, y) for the associated graphs G_+ (Hᵣ) 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 (Lᵣ, t) for a knot or link Lᵣ 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.

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