On diagrammatic bounds of knot volumes and spectral invariants

TL;DR

This paper investigates the limitations of diagrammatic measures like twist number in bounding hyperbolic knot volumes and spectral invariants, introducing double coil knots to establish new bounds and extending existing results on knot complement families.

Contribution

It demonstrates that twist numbers cannot generally provide two-sided bounds on volume or spectral invariants, and introduces double coil knots as a new family with such bounds, extending prior results.

Findings

Double coil knots admit two-sided bounds on volume and spectral invariants.

Neither twist number nor generalized twist number can bound volume or λ₁ for all knots.

A collection of double coil knot complements forms an expanding family iff their volume is bounded.

Abstract

In recent years, several families of hyperbolic knots have been shown to have both volume and $\lambda_1$ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized twist number of a diagram can provide two-sided bounds on either the volume or $\lambda_1$. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on $\lambda_1$. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded.