Workshop on Topics in Dehn Surgery
University of Texas at Austin
April 20-22, 2012


Ken Baker

On Dehn Surgeries between S1 x S2 and Lens Spaces

We consider knots in S1 x S2 with a Dehn surgery to a lens space and how this parallels what happens for knots in S3. The Cyclic Surgery Theorem enables a classification for non-longitudinal surgeries, while Berge's families of doubly primitive knots in S3 motivate our conjectural classification for longitudinal surgeries. We will compare this conjectural picture with the Berge Conjecture and propose a means of relating the two.

John Berge

The hyperbolic P/P, P/SFd, and P/SFm knots in S3

A knot k in S3 is P/P, P/SFd or P/SFm if k lies in a genus two Heegaard surface Σ of S3 bounding handlebodies H and H', adding a 2-handle to H' along k yields a solid torus, and adding a 2-handle to H along k yields a solid torus, an orientable Seifert fibered space over D2, or an orientable Seifert fibered space over the Mobius band respectively. The complete list of the hyperbolic P/P, P/SFd and P/SFm knots in S3, describes embeddings of each of these knots in S3, provides the surface slope of the exceptional surgery on each knot that yields a Seifert fibered space, and provides the indices of any exceptional fibers in the resulting Seifert fibered space. This is joint work with Sungmo Kang, with some assistance from Brandy Guntel. We intend to give an overview of the results and methods used to compile the list.

Radu Cebanu

Knots in lens spaces having S1 x S2 surgeries

We use Heegaard-Floer homology techniques to investigate properties of knots in lens spaces with a longitudinal S1 x S2 surgeries. We prove that such a knot must have a planar Seifert surface and we'll give further restrictions on its Knot Floer Homology.

Josh Greene

The realization problem

We will discuss the solution of the realization problem, which asks for the lens spaces that are realizable by integer surgery along a knot in the three-sphere.  We will begin by deriving a combinatorial obstruction to realizability from Heegaard Floer homology.  Then we will explore the combinatorial problem that results, involving changemakers and lattice embeddings.  We hope to give a fairly complete sketch of the determination of which connected sums of two or more lens spaces are realizable, and then discuss the extra issues that arise in handling the case of a single lens space.

Matt Hedden

An overview of Floer homology's perspective on the Berge conjecture

I'll survey the information which Floer homology provides about knots  admitting lens space surgeries, and the information it provides about knots in lens spaces with 3-sphere surgeries. I'll also focus a bit on recent work of Olga Plamenevskaya and myself that gives some contact geometric structure on the latter problem. Ample speculation about the ability of Floer homology to answer questions in the vein of the Berge conjecture will be provided.

Yi Ni

Dehn surgeries on knots in product manifolds

Given a knot in the product of a surface with an interval, when does a nontrivial surgery on the knot yield the same product 3-manifold? Obviously, if the knot has either a 0-crossing projection or a 1-crossing projection, then there exist such surgeries. We will prove that these are the only cases.

Tim Perutz

Cutting along tori in Heegaard Floer theory

Heegaard Floer homology behaves as a TQFT under cutting 3-manifolds along surfaces. In the "hat" specialization of Heegaard Floer homology, this behavior is established by the bordered Floer theory of Lipshitz-Ozsvath-Thurston. Joint work in progress with Lekili constructs a TQFT extension of the full Heegaard package, and has the bordered theory as a combinatorial shadow. This theory permits clean and sharp construction of Dehn surgery exact sequences, in which the three-fold cyclic symmetry of these sequences is a reflection of Jacobi's triple product identity from the theory of theta-functions. A more challenging question is what the extended theory tells us about the Heegaard Floer homology of 3-manifolds which split along an incompressible torus. I'll explain our exploratory work in this direction, which brings to bear tools from the representation theory of associative algebras.

Genevieve Walsh

Right-angled Coxeter groups and acute triangulations.

A triangulation of S2 yields a right-angled Coxeter group whose defining graph is the one-skeleton of that triangulation. In this case, the Coxeter group is the orbifold-fundamental group of a reflection orbifold which is finitely covered by a 3-manifold. We investigate the relationship between acute triangulations of S2 and the geometry of the associated right-angled Coxeter group. We will, of course, also discuss Dehn surgery and torus decompositions in this setting.