6 research outputs found
The limit set of the handlebody set has measure zero
This note fixes a small gap in Kerckhoff's proof that the limit set of the
handlebody set has measure zero.Comment: 3 page
The maximal tubes under the deformations of a class of 3-dimensional hyperbolic cone-manifolds
Recently, 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
Dehn surgeries along the Whitehead link complements. The basic method is based
on explicit holonomy computations using the A-polynomials and finding the
maximal tubes. One of the key tool is the Taylor expression of a geometric
component of the zero set of the A-polynomial in terms of the cone-angles. We
also show a sequence of Taylor expressions for Dehn surgered manifolds
converges to one for the limit hyperbolic manifold.Comment: 27 pages, 10 figure