arXiv subjects

Christopher K. Atkinson

Publications and source records attributed to Christopher K. Atkinson.

6 recordsLinked to original sources

arXiv2017

Guts and volume for hyperbolic $3$-orbifolds with underlying space $S^3$

For a hyperbolic $3$-orbifold with underlying space the $3$-sphere, we obtain a lower bound on its volume in the case that it contains an essential $2$-suborbifold with underlying space the $2$-sphere with four cone points. Our techniques involve computing the guts of the orbifold split along the $2$-suborbifold via a careful analysis of its topology. We also characterize the orbifolds of this type that have empty guts.

math.GT↗

arXiv2016

The lowest volume 3-orbifolds with high torsion

For each natural number n >= 4, we determine the unique lowest volume hyperbolic 3-orbifold whose torsion orders are bounded below by n. This lowest volume orbifold has base space the 3-sphere and singular locus the figure-8 knot, marked n. We apply this result to give sharp lower bounds on the volume of a hyperbolic manifold in terms of the order of elements in its symmetry group.

math.GT↗

arXiv2013

Small volume link orbifolds

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.

math.GT↗

arXiv2012

The smallest Haken hyperbolic polyhedra

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.

math.GT↗

arXiv2010

Two-sided combinatorial volume bounds for non-obtuse hyperbolic polyhedra

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, Schläfli's formula, and previous results of the author giving volume bounds for right-angled hyperbolic polyhedra.

math.GT↗

arXiv2010

Volume estimates for equiangular hyperbolic Coxeter polyhedra

An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where all dihedral angles are equal to π/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.

math.GT↗