On diagrammatic bounds of knot volumes and spectral invariants

Details On diagrammatic bounds of knot volumes and spectral invariants On diagrammatic bounds of knot volumes and spectral invariants

2009-01-02

arxiv.org/abs/0901.0119v1

Geometriae Dedicata 147 (2010):115-130

Mathematics

Geometric Topology

16 pages, 7 figures

Scientific paper

10.1007/s10711-009-9442-6

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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