University of Virginia Geometry Seminar 2012-13

Though many classical knot invariants are unchanged by Conway mutation, it has been known for some time that bigraded knot Floer homology distinguishes some knots from their Conway mutants. Similarly bigraded integral Khovanov homology detects some Conway mutations of links. However, in all known examples, the total rank of each of these homology theories is unchanged by Conway mutation. In joint work with Allison Moore, we consider an infinite family of knot pairs, where each pair is related by a genus 2 mutation (a generalization of Conway mutation). Each knot is distinguished from its mutant pair by both knot Floer and Khovanov homologies as bigraded theories, but the rank of these homologies is constant. This provides positive evidence towards a conjecture that the rank of knot Floer and Khovanov homologies is invariant under genus 2 mutation.

We find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of the sutured manifolds obtained by cutting the knot complements along the Seifert surfaces. Our examples provide the first use of sutured Floer homology, and not merely its Euler char- acteristic (a classical torsion), to distinguish Seifert surfaces.

By now there exist many categorifications of quantum link invariants, but as yet none of their ``colored'' versions (necessary to define an associated 3-manifold invariant) are functorial under link cobordisms. In this talk I propose a categorification of the colored Jones polynomial which is likely to be functorial. The construction uses a new, quasi-idempotent chain complex which categorifies a multiple of the Jones-Wenzl projector.

We use bordered Floer homology to study 3-manifolds obtained by gluing together two knot complements (gluing meridian to longitude). If the knots are non-trivial knots in the 3-sphere, we show that the Heegaard Floer homology of the resulting manifold has rank greater than one. By extending this approach to knots in arbitrary homology spheres, we hope to prove that a manifold whose Heegaard Floer homology has rank one cannot contain an incompressible torus. This is joint work with Matt Hedden.

Ozsvath and Szabo used Heegaard Floer homology to define an invariant of contact structures, where a non-vanishing invariant implies tightness. Using a surgery exact triangle for Heegaard Floer homology, Lisca and Stipsicz proved the existence of positive, tight contact structures on 3-manifolds resulted from contact surgeries on certain knots in the 3-sphere. After reviewing defintions and basic tools, I will outline the proof of this result.

The goal of this talk is to study geography and classification problem for Stein fillings of contact structures supported by planar open books. In the first part we will prove that for contact structures supported by planar open books Stein fillings have a finite geography. In the second part we will talk about classification of Stein fillings of virtually overtwisted contact structures on lens space L(pn+p+1,n+1).

In this talk, I describe a way from geometric analysis to try to prove the Hopf conjectures on positively curved manifolds. Namely, any compact even dimensional Riemannian manifold of positive sectional curvature has positive Euler characteristic, and S^2 * S^2 can not admit any Riemannian metric of positive sectional curvature. Our method may also give an affirmative answer to a longstanding problem of Yau: does there exist an effective circle action on a compact Riemannian manifold of positive sectional curvature? This is a joint work with Kefeng Liu.

In this talk, I will define a new connection which we call the canonical connection associated to every contact triad. We use it to study the contact instanton which is a generalization of Hofer's pseudo-holomorphic curve in symplectization. I will sketch how to use this connection to study the asymptotic behavior of contact instanton and also other analytic properties. We expect to study the moduli space of such contact instantons and then to define a new type of contact homology without involving symplectization. This is a joint work with Yong-Geun Oh.

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain
conditions on the fundamental group of M, I will construct quasi-isometric embeddings of either free Abelian or direct products of
non-Abelian free groups into the group of volume-preserving diffeomorphisms of M equipped with the L^p metric induced by a Riemannian metric on M. If time permits I will explain a relation between quasi-morphisms, the L^p metrics and quasi-isometric embeddings of vector spaces into the above group.

An easily stated question in knot theory asks when a crossing change in a knot diagram leaves the underlying knot unchanged. We will explore a related question involving cosmetic generalized crossing changes, which are defined in terms of Dehn surgery in the knot complement. In particular, we will find obstructions to the existence of cosmetic crossing changes in families of knots obtained from satellite and twisting operations on the torus.

Recently Sarkar defined maps for the combinatorial version of knot Floer homology (HFK) whose underlying grid maps could be viewed as births, deaths and saddles. Juhasz has shown that the hat version of HFK is functorial with respect to smooth decorated cobordisms. This leads one to ask the question, "Do these grid diagram maps induce maps on HFK that are invariant with respect to smooth marked isotopy classes of surfaces?" In this talk I will discuss and provide the necessary topological constructions to make sense of this question. Specifically, I will introduce grid movies and show how they correspond to smooth embedded surfaces and I will generalize Carter, Rieger and Saito's movie move theorem: to grid movies; smooth marked movies; and grid marked movies. If there is time, I will briefly sketch some of the necessary steps to answer the motivating question in the affirmative.

Motivated by the construction of the isomorphisms between Heegaard Floer and Seiberg--Witten Floer homologies (joint with Yi-Jen Lee and Clifford H. Taubes), we will describe a (Z+Z)-filtered monopole knot homology isomorphic to Ozsvath-Szabo's knot Floer homology.