Spectral geometry,link complements and surgery diagrams |
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Authors: | Marc Lackenby |
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Institution: | 1.Mathematical Institute,University of Oxford,Oxford,UK |
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Abstract: | We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold
M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of hyperbolic alternating link complements is expanding if and only
if they have bounded volume. We also provide examples of hyperbolic 3-manifolds which require ‘complicated’ surgery diagrams,
thereby proving that a recent theorem of Constantino and Thurston is sharp. Along the way, we find a new upper bound on the
bridge number of a knot that is not tangle composite, in terms of the twist number of any diagram of the knot. The proofs
rely on a theorem of Lipton and Tarjan on planar graphs, and also the relationship between many different notions of width
for knots and 3-manifolds. |
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Keywords: | |
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