The Determinant and Volume of 2-Bridge Links and Alternating 3-Braids

Abstract

We examine the conjecture, due to Champanerkar, Kofman, and Purcell [4] that vol(K) < 2π log det(K) for alternating hyperbolic links, where vol(K) = vol(S 3 \K) is the hyperbolic volume and det(K) is the determinant of K. We prove that the conjecture holds for 2-bridge links, alternating 3-braids, and various other infinite families. We show the conjecture holds for highly twisted links and quantify this by showing the conjecture holds when the crossing number of K exceeds some function of the twist number of K.