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

Abstract

We examine the conjecture, due to Champanerkar, Kofman, and Purcell that $\text{vol}(K)<2\pi \log \det (K)$ for alternating hyperbolic links, where $\text{vol}(K)=\text{vol}(S^3\backslash K)$ 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 $K$.

References in corpus (1)

• Volume bounds for weaving knots

Cited by in corpus (1)

• Mahler Measure and the Vol-Det Conjecture