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

TL;DR

This paper proves a conjecture relating hyperbolic volume and determinant for specific classes of alternating links, including 2-bridge links and alternating 3-braids, and explores conditions under which it holds.

Contribution

It establishes the conjecture for 2-bridge links, alternating 3-braids, and other families, and analyzes the influence of crossing and twist numbers on the conjecture's validity.

Findings

The conjecture holds for 2-bridge links.

The conjecture holds for alternating 3-braids.

High crossing number links satisfy the conjecture.

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 $K$.