Georgia Institute of Technology College of Sciences

Frohman and Gelca showed that the Kauffman bracket skein module of the thickened torus is the Z_2 invariant subalgebra A'_q of the quantum torus A_q. This shows that the Kauffman bracket skein module of a knot complement is a module over A'_q. We discuss a conjecture that this module is naturally a module over the double affine Hecke algebra H, which is a 3-parameter family of algebras which specializes to A'_q. We give some evidence for this conjecture and then discuss some corollaries. If time permits we will also discuss a related topological construction of a 2-parameter family of H-modules associated to a knot in S^3. (All results in this talk are joint with Yuri Berest.)

The notion of distance for a Heegaard splitting of athree-dimensional manifold $M$, introduced by John Hempel, has provedto be a very powerful tool for understanding the geometry and topologyof $M$. I will describe how distance, and a slight generalizationknown as subsurface projection distance, can be used to explore theconnection between geometry and topology at the center of the moderntheory hyperbolic three-manifolds.In particular, Schalremann-Tomova showed that if a Heegaard splittingfor $M$ has high distance then it will be the only irreducibleHeegaard splitting of $M$ with genus less than a certain bound. I willexplain this result in terms of both a geometric proof and atopological proof. Then, using the notion of subsurface distance, Iwill describe a construction of a manifold with multiple distinctlow-distance Heegaard splittings of the same (small) genus, and amanifold with both a high distance, low-genus Heegaard splitting and adistinct, irreducible high-genus, low-distance Heegaard splitting.

The $4$-genus of a knot is an important measure of complexity, related tothe unknotting number. A fundamental result used to study the $4$-genusand related invariants of homology classes is the Thom conjecture,proved by Kronheimer-Mrowka, and its symplectic extension due toOzsvath-Szabo, which say that closed symplectic surfacesminimize genus.Suppose (X, \omega) is a symplectic 4-manifold with contact type bounday and Sigma is a symplectic surface in X such that its boundary is a transverse knot in the boundary of X. In this talk we show that there is a closed symplectic 4-manifold Y with a closed symplectic submanifold S such that the pair (X, \Sigma) embeds symplectically into (Y, S). This gives a proof of the relative version of Symplectic Thom Conjecture. We use this to study 4-genus of fibered knots in the 3-sphere.We will also discuss a relative version of Giroux's criterion of Stein fillability. This is joint work with Siddhartha Gadgil