Publications (4)0 Total impact
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ABSTRACT: This paper proves lower bounds on the volume of a hyperbolic 3-orbifold whose
singular locus is a link. We identify the unique smallest volume orbifold whose
singular locus is a knot or link in the 3-sphere, or more generally in a Z_6
homology sphere. We also prove more general lower bounds under mild homological
hypotheses.
11/2012;
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ABSTRACT: We determine the lowest volume hyperbolic Coxeter polyhedron whose
corresponding hyperbolic polyhedral 3-orbifold contains an essential
2-suborbifold, up to a canonical decomposition along essential hyperbolic
triangle 2-suborbifolds.
08/2011;
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Christopher K. Atkinson
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ABSTRACT: We give a method for computing upper and lower bounds for the volume of a
non-obtuse hyperbolic polyhedron in terms of the combinatorics of the
1-skeleton. We introduce an algorithm that detects the geometric decomposition
of good 3-orbifolds with planar singular locus and underlying manifold the
3-sphere. The volume bounds follow from techniques related to the proof of
Thurston's Orbifold Theorem, Schl\"afli's formula, and previous results of the
author giving volume bounds for right-angled hyperbolic polyhedra.
08/2010;
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Christopher K. Atkinson
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ABSTRACT: An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where all dihedral angles are equal to \pi/n for some fixed integer n at least 2. It is a consequence of Andreev's theorem that either n=3 and the polyhedron has all ideal vertices or that n=2. Volume estimates are given for all equiangular hyperbolic Coxeter polyhedra. Comment: 27 pages, 11 figures; corrected typo in Theorem 2.4
04/2008;
