The Vol-Det Conjecture for highly twisted alternating links

TL;DR

This paper advances the understanding of the Vol-Det Conjecture by improving bounds on the number of twists for which it holds, specifically for highly twisted alternating links, and refines related inequalities for volumes and determinants.

Contribution

It improves Burton's bounds on crossings and Stoimenow's inequalities for highly twisted alternating and arborescent links, extending the conjecture's verified cases.

Findings

Enhanced bounds for the number of twists in the Vol-Det Conjecture.

Refined inequalities relating hyperbolic volume and determinant.

Extended validity of the conjecture to more complex links.

Abstract

The Vol-Det Conjecture, formulated by Champanerkar, Kofman and Purcell, states that there exists a specific inequality connecting the hyperbolic volume of an alternating link and its determinant. Among the classes of links for which this conjecture holds are all alternating hyperbolic knots with at most 16 crossings, 2-bridge links, and links that are closures of 3-strand braids. In the present paper, Burton's bound on the number of crossings for which the Vol-Det Conjecture holds is improved for links with more than eight twists. In addition, Stoimenow's inequalities between hyperbolic volumes and determinants are improved for alternating and alternating arborescent links with more than eight twists.