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 $-1$), 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.
