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

2017/12/13 by Wenzhao Chen, Chen, Wenzhao
Mathematics · Medicine · #Geometric and Algebraic Topology #Bone health and treatments #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1712.04993

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.

Related

• Twisted Alexander polynomials of 2-bridge knots
• On Alexander-Conway polynomials of two-bridge links
• On zeros of the Alexander polynomial of an alternating knot
• Alexander Polynomials of closed alternating braids
• On Fox's trapezoidal conjecture