On the Alexander polynomial and the signature invariant of two-bridge knots

TL;DR

This paper proves the Hirasawa-Murasugi conjecture linking the stable coefficients of the Alexander polynomial to the signature invariant specifically for two-bridge knots, confirming a key relationship in knot theory.

Contribution

It establishes the conjecture for two-bridge knots, advancing understanding of the Alexander polynomial's structure and its relation to knot invariants.

Findings

Confirmed the conjecture for two-bridge knots

Established a link between stable coefficients and signature invariant

Enhanced understanding of Alexander polynomial properties

Abstract

Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa-Murasugi conjecture for two-bridge knots.