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

Cusp types of quotients of hyperbolic knot complements

This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, $S^2(2,4,4)$ cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orbifold with a $S^2(2,3,6)$ cusp, it also covers an orbifold with a $S^2(3,3,3)$ cusp.Comment: 11 pages, 4 figures, updates to this version include a complete classification of cusps of non-orientable quotients, minor tweaks to the exposition, an update to the proof of Theorem 1.1 and the removal of a discussion about the uniqueness of rigid cusp quotients of knot complemen

Cusp types of quotients of hyperbolic knot complements

This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, S2(2,4,4) cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orbifold with a S2(2,3,6) cusp, it also covers an orbifold with a S2(3,3,3) cusp. We end with a discussion that shows all cusp types arise in the quotients of link complements.Mathematic

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.Comment: 19 pages, 2 figures. v2: minor changes to intro. v3: accepted version, to appear in Math. Res. Letter

Geometry of planar surfaces and exceptional fillings

Mathematic

On the complexity of cusped non-hyperbolicity

We show that the problem of showing that a cusped 3-manifold M is not hyperbolic is in NP, assuming S3-RECOGNITION is in coNP. To this end, we show that IRREDUCIBLE TOROIDAL RECOGNITION lies in NP. Along the way we unconditionally recover SATELLITE KNOT RECOGNITION lying in NP. This was previously known only assuming the Generalized Riemann Hypothesis. Our key contribution is to certify closed essential normal surfaces as essential in polynomial time in compact orientable irreducible ∂-irreducible triangulations. Our work is made possible by recent work of Lackenby showing several basic decision problems in 3-manifold topology are in NP or coNP.Mathematic

Exceptional surgeries in 3-manifolds

Myers shows that every compact, connected, orientable 3--manifold with no 2--sphere boundary components contains a hyperbolic knot. We use work of Ikeda with an observation of Adams-Reid to show that every 3--manifold subject to the above conditions contains a hyperbolic knot which admits a non-trivial non-hyperbolic surgery, a toroidal surgery in particular. We conclude with a question and a conjecture about reducible surgeries.Mathematic