Neil R. Hoffman

Verified Computations for Hyperbolic 3-Manifolds

For a given cusped 3-manifold M admitting an ideal triangulation, we describe a method to rigorously prove that either M or a filling of M admits a complete hyperbolic structure via verified computer calculations. Central to our method is an implementation of interval arithmetic and Krawczyk’s test. These techniques represent an improvement over existing algorithms as they are faster while accounting for error accumulation in a more direct and user-friendly way.

Commensurability classes containing three knot complements

This paper exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes.

Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling

An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double branched covers of links in S^3. We prove the existence of infinitely many such examples (in several distinct families) using a mix of hyperbolic geometry, Floer theory, and verified computer calculations. Of independent interest is our technique for using interval arithmetic to certify symmetry groups and non-existence of isometries of cusped hyperbolic 3-manifolds. In the process, we give examples of 1-cusped hyperbolic 3-manifolds of Heegaard genus 3 with two distinct lens space fillings. These are the first examples where multiple Dehn fillings drop the Heegaard genus by more than one, which answers a question of Gordon.

Small knot complements, exceptional surgeries and hidden symmetries

This paper provides two obstructions to small knot complements in S 3 admitting hidden symmetries. The first obstruction is being cyclically commensurable with another knot complement. This result provides a partial answer to a conjecture of Boileau, Boyer, Cebanu and Walsh. We also provide a second obstruction to admitting hidden symmetries in the case where a small knot complement covers a manifold admitting some symmetry and at least two exceptional surgeries. 57M12, 57M25; 57M10

Geometry of planar surfaces and exceptional fillings

If a hyperbolic 3-manifold admits an exceptional Dehn filling, then the length of the slope of that Dehn filling is known to be at most six. However, the bound of six appears to be sharp only in the toroidal case. In this paper, we investigate slope lengths of other exceptional fillings. We construct hyperbolic 3-manifolds that have the longest known slopes for reducible fillings. As an intermediate step, we show that the problem of finding the longest such slope is equivalent to a problem on the maximal density horoball packings of planar surfaces, which should be of independent interest. We also discuss lengths of slopes of other exceptional Dehn fillings, and prove that six is not realized by a slope corresponding to a small Seifert fibered space filling.

Double bubbles inS 3 andH 3

AbstractWe prove the double bubble conjecture in the three-sphereS3 and hyperbolic three-spaceH3 in the cases where we can apply Hutchings theory:• InS3, when each enclosed volume and the complement occupy at least 10% of the volume ofS3.• inH3, when the smaller volume is at least 85% that of the larger. A balancing argument and asymptotic analysis reduce the problem inS3 andH3 to some computer checking. The computer analysis has been designed and fully implemented for both spaces.