Workshop on Topics in Dehn Surgery


Ken BakerOn Dehn Surgeries between S^{1} x S^{2} and Lens SpacesWe consider knots in S^{1} x S^{2} with a Dehn surgery to a lens space and how this parallels what happens for knots in S^{3}. The Cyclic Surgery Theorem enables a classification for nonlongitudinal surgeries, while Berge's families of doubly primitive knots in S^{3} 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 BergeThe hyperbolic P/P, P/SF_{d}, and P/SF_{m} knots in S^{3}A knot k in S^{3} is P/P, P/SF_{d} or P/SF_{m} if k lies in a genus two Heegaard surface Σ of S^{3} bounding handlebodies H and H', adding a 2handle to H' along k yields a solid torus, and adding a 2handle to H along k yields a solid torus, an orientable Seifert fibered space over D^{2}, or an orientable Seifert fibered space over the Mobius band respectively. The complete list of the hyperbolic P/P, P/SF_{d} and P/SF_{m} knots in S^{3}, describes embeddings of each of these knots in S^{3,} 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 CebanuKnots in lens spaces having S^{1} x S^{2} surgeriesWe use HeegaardFloer homology techniques to investigate properties of knots in lens spaces with a longitudinal S^{1} x S^{2} 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 GreeneThe realization problemWe 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 threesphere. 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 HeddenAn overview of Floer homology's perspective on the Berge conjectureI'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 3sphere 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 NiDehn surgeries on knots in product manifoldsGiven a knot in the product of a surface with an interval, when does a nontrivial surgery on the knot yield the same product 3manifold? Obviously, if the knot has either a 0crossing projection or a 1crossing projection, then there exist such surgeries. We will prove that these are the only cases. Tim PerutzCutting along tori in Heegaard Floer theoryHeegaard Floer homology behaves as a TQFT under cutting 3manifolds along surfaces. In the "hat" specialization of Heegaard Floer homology, this behavior is established by the bordered Floer theory of LipshitzOzsvathThurston. 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 threefold cyclic symmetry of these sequences is a reflection of Jacobi's triple product identity from the theory of thetafunctions. A more challenging question is what the extended theory tells us about the Heegaard Floer homology of 3manifolds 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 WalshRightangled Coxeter groups and acute triangulations.A triangulation of S^{2} yields a rightangled Coxeter group whose defining graph is the oneskeleton of that triangulation. In this case, the Coxeter group is the orbifoldfundamental group of a reflection orbifold which is finitely covered by a 3manifold. We investigate the relationship between acute triangulations of S^{2} and the geometry of the associated rightangled Coxeter group. We will, of course, also discuss Dehn surgery and torus decompositions in this setting.
