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.