Volume and Determinant Densities of Hyperbolic Rational Links

TL;DR

This paper investigates the properties and distributions of volume and determinant densities in hyperbolic rational links, demonstrating their density in a specific interval and constructing sequences with converging densities.

Contribution

It introduces new constructions of alternating knots with converging densities and analyzes the distribution and bounds of these invariants for hyperbolic rational links.

Findings

Volume and determinant densities are dense in [0, v_{oct}]

Constructed sequences of alternating knots with converging densities

Established upper bounds and distribution results for these invariants

Abstract

The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval [0,v_{oct}]. We construct sequences of alternating knots whose volume and determinant densities both converge to any x in [0,v_{oct}]. We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these invariants.