27 research outputs found
On diagrammatic bounds of knot volumes and spectral invariants
In recent years, several families of hyperbolic knots have been shown to have
both volume and (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 . 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 . 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.Comment: 16 pages, 7 figure
Counting and effective rigidity in algebra and geometry
The purpose of this article is to produce effective versions of some rigidity
results in algebra and geometry. On the geometric side, we focus on the
spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic
hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum
determines the commensurability class of the 2-manifold (resp., 3-manifold). We
establish effective versions of these rigidity results by ensuring that, for
two incommensurable arithmetic manifolds of bounded volume, the length sets
(resp., the complex length sets) must disagree for a length that can be
explicitly bounded as a function of volume. We also prove an effective version
of a similar rigidity result established by the second author with Reid on a
surface analog of the length spectrum for hyperbolic 3-manifolds. These
effective results have corresponding algebraic analogs involving maximal
subfields and quaternion subalgebras of quaternion algebras. To prove these
effective rigidity results, we establish results on the asymptotic behavior of
certain algebraic and geometric counting functions which are of independent
interest.Comment: v.2, 39 pages. To appear in Invent. Mat
On diagrammatic bounds of knot volumes and spectral invariants, Geom. Dedicata 147
Abstract In recent years, several families of hyperbolic knots have been shown to have both volume and λ 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 λ 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 λ 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