Zeros of the Jones Polynomial for Multiple Crossing-Twisted Links

Abstract

Let D be a general connected reduced alternating link diagram, C be the set of crossings of D and C′ be the nonempty subset of C. In this paper we first define a multiple crossing-twisted link family {Dn(C′)|n=1,2,…} based on D and C′, which produces (2,2n+1)-torus knot family, the link family An defined in Chang and Shrock (Physica A 301:196–218, 2001) and the pretzel link family P(n,n,n) as special cases. Then by applying Beraha-Kahane-Weiss’s Theorem we prove that limits of zeros of Jones polynomials of {Dn(C′)|n=1,2,…} are the unit circle |z|=1 (It is independent of the selections of D and C′) and several isolated limits, which can be determined by computing flow polynomials of subgraphs of G corresponding to D. Furthermore, we use the method of Brown and Hickman (Discrete Math. 242:17–30, 2002) to prove that, for any e>0, all zeros of Jones polynomial of the link Dn(C) lie inside the circle |z|=1+e, provided that n is large enough. Our results extend results of F.Y. Wu, J. Wang, S.-C. Chang, R. Shrock and the present authors and refine partial result of A. Champanerkar and L. Kofman.