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 λ1 (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 λ1. 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 λ1. 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. © 2009 Springer Science+Business Media B.V.