Geometrically and diagrammatically maximal knots

TL;DR

This paper investigates which knots nearly maximize the ratio of volume and determinant to crossing number, showing that many alternating knots and links achieve these near-maximal ratios, thus advancing understanding of geometric and algebraic knot invariants.

Contribution

The paper identifies families of alternating knots and links that simultaneously nearly maximize both volume-to-crossing and determinant-to-crossing ratios.

Findings

Many alternating knots and links nearly maximize both ratios.

The ratios are bounded above by the volume of a regular ideal octahedron.

Results support conjectures relating geometric and algebraic knot invariants.

Abstract

The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We show that many families of alternating knots and links simultaneously maximize both ratios.