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

2017/04/07 by Burton, Stephan D.
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1704.02344

Abstract

We examine the conjecture, due to Champanerkar, Kofman, and Purcell that vol(K) < 2 πlog det (K) for alternating hyperbolic links, where vol(K) = vol(S³\backslash 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.

Related

• The Vol-Det Conjecture for highly twisted alternating links
• Twist Number and the Alternating Volume of Knots
• The volume of hyperbolic alternating link complements
• Generating Infinitely Many Hyperbolic Knots with Plats
• Volume and Determinant Densities of Hyperbolic Rational Links