This is the website for the joint topology seminar between Duke, NCSU and UNC. The seminar will normally meet on selected Tuesdays with the location rotating between Duke, NCSU and UNC. An informal pretalk seminar aimed at graduate students will also be held before some talks.
Please contact one of the organizers to be added to our mailing list.
(Louisiana State University)
|Jan 23 4:30pm||NCSU
|Combinatorial invariants of transverse links via cyclic branched covers||Adam Saltz
(University of Georgia)
|Mar 26 3:15pm||Duke
|Link homology and Floer homology in pictures by cobordisms||Melissa Zhang
|Apr 2 3:15pm||Duke
|Annular Khovanov homology and 2-periodic links|
|Apr 9 3:15pm||Duke
|Trivial tangles, compressible surfaces and Floer homology|
Combinatorial invariants of transverse links via cyclic branched covers
Abstract: Grid homology is a version of knot Floer homology in the 3-sphere that is entirely combinatorial and simple to define. Exploiting this, Ozsvath, Szabo, and Thurston defined a combinatorial invariant of transverse links in the 3-sphere using grid homology, which was then used to show that certain knot types are transversely non-simple by Ng, Ozsvath, and Thurston. This is particularly interesting because there are few invariants well suited for such purpose. More generally, there is also an invariant for transverse links in an arbitrary 3-manifold defined by Lisca, Ozsvath, Stipsicz, and Szabo, using an open-book decomposition. However, this invariant is difficult to compute in general. In this talk, for a transverse link in the 3-sphere, we will define combinatorial invariants by considering its lifts in its cyclic branched covers, using Levine's grid-like diagrams for knot Floer homology, and show that they coincide with the LOSS invariants for the lifts. This is joint work with Shea Vela-Vick.
Link homology and Floer homology in pictures by cobordisms
Abstract: There are no fewer than eight link homology theories which admit spectral sequences from Khovanov homology. These theories have very different origins -- representation theory, gauge theory, symplectic topology -- so it's natural to ask for some kind of unifying theory. I will attempt to describe this theory using Bar-Natan's pictorial formulation of link homology. This strengthens a result of Baldwin, Hedden, and Lobb and proves new functoriality results for several link homology theories. I won't assume much specific knowledge of these link homology theories, and the bulk of the talk will be accessible to graduate students!
Annular Khovanov homology and 2-periodic links
Abstract: I will exhibit a spectral sequence from the annular Khovanov homology of a 2-periodic link to that of its quotient, which in turn proves rank inequalities and decategorifies to polynomial congruences. While previous work used heavier algebraic machinery to prove this rank inequality in a particular sl_2 weight space grading, we instead mimic Borel's construction of equivariant cohomology and employ grading considerations to give a combinatorial proof of the rank inequality for all quantum and sl_2 weight space gradings. Curiously, the same methods suggest a similar spectral sequence relating the Khovanov homology of a 2-periodic link and the annular Khovanov homology of its quotient link. We'll discuss partial results on this front.
Trivial tangles, compressible surfaces and Floer homology
Abstract: Heegaard Floer homology has different extensions for 3-manifolds with boundary. In this talk, we will recall some basics of these extensions and explain how they can be used to give a computationally effective way for detecting boundary parallel components of tangles, and existence of homologically essential compressing disks. The fact that these are checkable by computer, is based on the factoring algorithm of Lipshitz-Ozsvath-Thurston for computing bordered Floer homology, and our extension of it to compute bordered-sutured Floer homology. This is a joint work with Robert Lipshitz