Maximal tubes under the deformations of 3-dimensional hyperbolic cone-manifolds

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Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-manifold is decreasing and that summed with the cone-angle squared is increasing as we deform the cone-angles. We confirm this near 0 cone-angles for an infinite family of hyperbolic cone-manifolds obtained by Deln surgeries along the Whitehead link complements. The basic method rests on explicit holonomy computations using the A-polynomials and finding the maximal tubes. One of the key tools is the Taylor expansion of a geometric component of the zero set of the A-polynomial in terms of the cone-angles. We also show that a sequence of Taylor expansions for Dehn surgered manifolds converges to 1 for the limit hyperbolic manifold.
Publisher
CONSULTANTS BUREAU/SPRINGER
Issue Date
2006-09
Language
English
Article Type
Article
Citation

SIBERIAN MATHEMATICAL JOURNAL, v.47, no.5, pp.955 - 974

ISSN
0037-4466
DOI
10.1007/s11202-006-0107-5
URI
http://hdl.handle.net/10203/7629
Appears in Collection
MA-Journal Papers(저널논문)
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