Mathematics > Geometric Topology
[Submitted on 2 Jan 2009]
Title:On diagrammatic bounds of knot volumes and spectral invariants
View PDFAbstract: 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.
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