The big Dehn surgery graph and the link of S^3

In a talk at the Cornell Topology Festival in 2005, W. Thurston discussed a graph which we call "The Big Dehn Surgery Graph", B. Here we explore this graph, particularly the link of S^3, and prove facts about the geometry and topology of B. We also investigate some interesting subgraphs and pose what we believe are important questions about B.Comment: 15 pages, 4 figures, 4 ancillary files. Reorganized and shortened from previous versions, while correcting one error in the proof of Theorem 5.4. Also, ancillary files detailing our computations with the computer program ORB have been provide

On knot complements that decompose into regular ideal dodecahedra

Aitchison and Rubinstein constructed two knot complements that can be decomposed into two regular ideal dodecahedra. This paper shows that these knot complements are the only knot complements that decompose into n regular ideal dodecahedra, providing a partial solution to a conjecture of Neumann and Reid.Mathematic

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.Mathematic

Small knot complements, exceptional surgeries, and hidden symmetries

This paper provides two obstructions to small knot complements in S3 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.Mathematic

On manifolds with multiple lens space filings

An irreducible 3--manifold with torus boundary either is a Seifert fibered space or admits at most three lens space fillings according to the Cyclic Surgery Theorem. We examine the sharpness of this theorem by classifying the non-hyperbolic manifolds with more than one lens space filling, classifying the hyperbolic manifolds obtained by filling of the Minimally Twisted 5 Chain complement that have three lens space fillings, showing that the doubly primitive knots in $S^3$ and $S^1 \times S^2$ have no unexpected extra lens space surgery, and showing that the Figure Eight Knot Sister Manifold is the only non-Seifert fibered manifold with a properly embedded essential once-punctured torus and three lens space fillings.Comment: 30 pages, 13 figure