Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in Z/2Z is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that…

During the last thirty years, several (families of) functionals have been defined which model self-avoidance: their values tend to infinity if an embedded object degenerates, e.g., if a sequence of closed simple curves converges to a curve with a self-intersection. Many of these functionals exhibit regularizing effects: they not only ensure embeddedness but in fact…

Let G denote a finite group and X a CW-complex. A homotopy group action is defined as a homotopy class of maps BG -- BAut(X). In this talk we will analyze these actions using techniques from group cohomology. We will show how they relate to geometric actions and how they can be used to construct…

Given a link diagram L, one can apply a Skein relation to each crossing to yield a cube of resolutions. These skein relations come from the braiding in the category of Uq(sln) representations. When n2, we have the Khovanov cube of resolutions with edge maps defined by (co)pants conordisms. We may then apply a smooth…

We give a brief overview of factorization homology theory due to Ayala, Francis and Tanaka and explain how it leads to a (still mostly conjectural) generalization of graph homology to homotopy commutative algebras, and an efficient computation of knot invariants coming from factorization homology (at least for alternating links).

Khovanov homology is a recent link invariant that lifts the Jones polynomial. We analyze torsion in Khovanov homology by describing a related homology theory that lifts the chromatic polynomial. In particular, we describe torsion in Khovanov homology of several link  families and compute the fourth extreme coefficients of the Jones polynomial for certain links.

In the 1980’s Furuta and Fintushel-Stern applied the theory of instantons and Chern-Simons invariants to develop a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. These results, together with some 4-dimensional constructions can be used to show that iterated Whitehead doubles of positive…

Dehn surgery is a fundamental operation in three-manifold topology which turns a knot into a new three-manifold. We characterize Dehn surgeries between certain lens spaces and relate this to an elementary question in knot theory. This is joint work with Allison Moore.

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of $S^1 \times S^3$. Specifically, we show that for any smoothly embedded 3-manifold Y representing a generator of H_3(X), a suitable version of the Heegaard Floer d invariant of Y, defined using twisted coefficients, is a diffeomorphism…

A large variety of multimedia data inference problems require analysis of repeated structures.  In audio, for instance, the rhythm, or ``pulse'' of the music, occurs in a periodic pattern, and understanding this pattern is an important preprocessing step in music information retrieval.  In medical video analysis, there is interest in determining heart pulse rate in…

In the 1960s Tutte observed that the value of the chromatic polynomial of planar triangulations at (golden ratio +1) obeys a number of remarkable properties. In this talk I will explain how TQFT gives rise to a conceptual framework for studying planar triangulations. I will discuss several extensions of Tutte's results and applications to the…

Earlier Helme Guizon and Rong have defined chromatic graph homology complex for a graded commutative algebra, and it is easy to extend the definition to graded commutative DG algebra. One of important applications, considered earlier in our joint work with Radmila Sazdanovic, is to the case of an algebra computing cohomology of a manifold, such…

In contact topology, invariants of Legendrian submanifolds in 1-jet spaces have been obtained through a variety of techniques. I will discuss how I am enriching one Morse-theoretic invariant, Generating Family Cohomology, to an A-infinity algebra by constructing product maps. The construction uses moduli spaces of Morse flow trees: spaces of intersecting gradient trajectories of functions…