On diagrammatic bounds of knot volumes and spectral invariants

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.

16 pages, 7 figures

References in corpus (1)

• Symmetric links and Conway sums: volume and Jones polynomial

Cited by in corpus (2)

• Generalized augmented alternating links and hyperbolic volume
• A survey of hyperbolic knot theory