On the Kashaev signature conjecture

David Cimasoni; Livio Ferretti

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arXiv · 2310.16729

On the Kashaev signature conjecture

David Cimasoni·Livio Ferretti

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Abstract

In 2018, Kashaev introduced a square matrix indexed by the regions of a link diagram, and conjectured that it provides a novel way of computing the Levine-Tristram signature and Alexander polynomial of the corresponding oriented link. In this article, we show that for the classical signature (i.e. the Levine-Tristram signature at -1), this conjecture follows from the seminal work of Gordon-Litherland. We also relate Kashaev's matrix to Kauffman's "Formal Knot Theory" model of the Alexander polynomial. As a consequence, we establish the Alexander polynomial and classical signature parts of the conjecture for arbitrary links, as well as the full conjecture for definite knots.

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David Cimasoni, Livio Ferretti. 2023-10-25. On the Kashaev signature conjecture. https://arxiv.org/abs/2310.16729

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