On the location of zeros of the Homfly polynomial

Abstract

Wu, Wang, Chang and Shrock initiated the study of zeros of the Jones polynomialsince it was the special case of partition functions of the Potts model in physics.The Homfly polynomial is the generalization of the Jones polynomial. LetL be an orientedlink, and PL(v, z) be its Homfly polynomial. In this paper, we study zeros ofPL(v, z) withz fixed. We prove the so-called unit-circle theorem for a family of generalized Jaeger’s links{Dn(G)|n = 1, 2,...} whichstates that |v| = 1 is the limit of zeros of Homfly polynomials of generalized Jaeger’s links{Dn(G)|n = 1, 2,...} ifG is bridgeless. Similar to the result of the Jones polynomial, we also prove that zeros ofHomfly polynomials are dense in the whole complex plane.