On the Kashaev signature conjecture
Volume 266 / 2024
Fundamenta Mathematicae 266 (2024), 275-287
MSC: Primary 57K10
DOI: 10.4064/fm231025-15-5
Published online: 10 July 2024
Abstract
In 2018, Kashaev introduced a square matrix indexed by the regions of a link diagram and conjectured that it provides a 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 ), this conjecture follows from the 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.
