Right-angled polyhedra and alternating links

TL;DR

This paper introduces a new geometric invariant called the right-angled volume for prime alternating links, computed via hyperbolic right-angled ideal polyhedra, providing a lower bound for the link's hyperbolic volume.

Contribution

It establishes a novel method to associate hyperbolic right-angled polyhedra with prime alternating links and defines a new volume invariant that bounds the hyperbolic volume from below.

Findings

The right-angled volume is a new geometric invariant for alternating links.

The right-angled volume provides a proven lower bound for the hyperbolic volume.

An explicit procedure for computing this invariant from link diagrams is given.

Abstract

To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume of the alternating link. We give an explicit procedure to compute the right-angled volume from any alternating link diagram, and prove that it is a new lower bound for the hyperbolic volume of the link.