A Proof of the Kashaev Signature Conjecture

Abstract

In 2018 Kashaev introduced a diagrammatic link invariant conjectured to be twice the Levine-Tristram signature. If true, the conjecture would provide a simple way of computing the Levine-Tristram signature of a link by taking the signature of a real symmetric matrix associated with a corresponding link diagram. This paper establishes the conjecture using the Seifert surface definition of the Levine-Tristram signature on the disjoint union of an oriented link and its reverse. The proof also reveals yet another formula for the Alexander polynomial.