Low-Dimensional Topology and Categorification


All talks will be in Room S-240, in the basement of the math department building.
Mikhail Khovanov, Diagrammatics for categorified quantum groups
Categorified quantum groups are monoidal categories and 2-categories described by carefully chosen relations on interacting decorated strings labelled by simple roots of the corresponding Lie algebras. We shall review some ideas behind this story in the context of several basic examples.
Charles Livingston, The nonorientable 4-genus of knots
The nonorientable 4-genus of a knot is the minimum first betti number of a nonorientable surface in the 4-ball having the knot as boundary. In this talk I will describe joint work with Pat Gilmer in which we develop new methods for bounding the nonorientable genus of a knot. Techniques range from classical methods, based on the knot signatures, the arf invariant, and linking forms, to applications of Casson-Gordon theory, to obstructions arising via Heegaard-Floer theory
Jacob Rasmussen, Handle decompositions and exotic S2xS2's
Akhmedov and Park have recently constucted an exotic S2xS2. I'll explain how knot Floer homology can be used to show that this manifold does not admit a handle decomposition composed of only 0,2, and 4-handles.
Liam Watson, L-spaces and left-orderability (No Video Available)
Examples suggest that there is a correspondence between L-spaces and 3-manifolds with fundamental group that cannot be left-ordered. This talk will introduce these notions, present certain families of examples, and discuss some related questions.
Aaron Lauda, Extended graphical calculus for categorified sl2
Igor Frenkel conjectured that the quantum enveloping algebra of sl2 could be categorified at generic q utilizing Lusztig's canonical basis. In this talk we will review a realization of this conjecture. This categorification is given by a 2-category that is the idempotent completion, or Karoubi envelope, of a 2-category defined using a graphical calculus. Here we introduce joint work with Khovanov, Mackaay, and Stosic defining an `extended' graphical calculus for the Karoubi envelope. This extended calculus can be used to prove the main categorification result over the integers and also reveals surprising relationships between closed diagrams in the sl2 calculus and the combinatorics of symmetric functions.
Anna Beliakova, A categorification of the Casimir of quantum sl2 (joint with A. Lauda and M. Khovanov)
Here we will give an explicit complex categorifying the Casimir, and show that it belongs to the center of the Khovanov-Lauda 2-category
Robert Lipshitz, Introduction to bordered Heegaard Floer homology
Bordered Floer homology is an extension of Ozsvath-Szabo's invariant HF-hat to 3-manifolds with boundary. After discussing the general structure of bordered Floer homology, and how it relates to the rest of the Heegaard Floer package, we will sketch how bordered Floer homology can be used to compute HF-hat of closed 3-manifolds and, time permitting, the maps induced by 4-dimensional cobordisms. This is joint work with Peter Ozsvath and Dylan Thurston
Dylan Thurston, A faithful action of the mapping class group from Heegaard Floer homology
Bordered Heegaard Floer homology extends the theory of Heegaard Floer invariants to 3-manifolds with surface boundary. In particular, it gives an action of the (strongly based) mapping class group of a surface on an appropriate category of modules. We show that this action is faithful, at least in the second-extremal grading. Indeed, the total rank of the homology is given by appropriate interesection numbers.
Tomasz Mrowka, Khovanov homology is an unknot detector
This talk will sketch a proof due to Kronheimer and the speaker that Khovanov homology can be understood as a page in a spectral sequence converging to a version of Instanton Floer homology. The later theory is defined for knots in arbitrary three manifolds and even sutured three manifolds. Thanks to this flexibility it can be shown to detect the unknot. A consequence of this is that Khovanov homology also detects the unknot.
Peter Ozsvath, Bordered Heegaard Floer Homology
Yi Ni, Dehn surgeries on knots in product manifolds
Let $K$ be a knot in a thickened surface $F\times I$. Suppose a nontrivial surgery on $K$ yields a manifold which is homeomorphic to $F\times I$, then the minimal projection of $K$ on $F$ has either zero or one crossing. I will discuss the proof, as well as some related questions about mapping class groups and cosmetic surgeries.
Scott Morrison, The blob complex
The blob complex is simultaneously a generalisation of an extended n-dimensional TQFT and (when n=1) Hochschild homology. I'll describe the framework and a gluing formula based on A_\infty tensor products. I'll then give a higher dimensional analogue of the Deligne conjecture, which provides an action of the little discs operad on Hochschild cohomology. Of interest, also, may be our (yet another) definition of an n-category.
Lenhard Ng, Knot contact homology and transverse knots
We describe an enhanced version of knot contact homology that yields invariants of topological knots and transverse knots in R3. In particular, this constitutes a surprisingly effective invariant of transverse knots, and we are able to show transverse nonsimplicity for several new knot types.
Louis Kauffman, Categorifications of the arrow polynomial for virtual knots
Two categorifications are given for the arrow polynomial, an extension of the Jones polynomial (using an oriented Kauffman bracket state summation) for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer (the arrow number) calculated from each loop in an oriented state summation for the bracket. The categorifications are based on new gradings associated with these arrow numbers, and give homology theories associated with oriented virtual knots and links via extra structure on the Khovanov chain complex. Applications are given to the estimation of virtual crossing number and surface genus of virtual knots and links. We give examples, discovered by Aaron Kaestner, of pairs of distinct knots with the same arrow polynomial, but different homologies. See arXiv:0906.3408, arXiv:0712.2546 and arXiv:0810.3858 for background for this talk.
Eli Grigsby, On Khovanov-Seidel quiver algebras and sutured Khovanov homology
Ozsvath-Szabo's spectral sequence from Khovanov homology to Heegaard Floer homology has generated a number of interesting applications to questions in low-dimensional topology. By combining the constructions of Ozsvath-Szabo with sutured manifold theory, we now have an enhanced understanding of the algebraic structure of the connection. In particular, a generalization of Juhasz's surface decomposition theorem implies that the algebra of the spectral sequence behaves ``as expected" under natural geometric operations like cutting and stacking.
In this talk, I will discuss joint work in progress with Stephan Wehrli aimed at understanding how the connection between Khovanov and Heegaard-Floer homology behaves under gluing. More precisely, we will see how to recover (a portion of) the sutured version of Khovanov homology as the Hochschild homology of certain bimodules over quiver algebras defined by Khovanov-Seidel. Along the way, we will discuss an intriguing relationship between these Khovanov-Seidel bimodules and certain bimodules appearing in the bordered Floer package of Lipshitz-Ozsvath-Thurston.
Sabin Cautis, D-modules and categorical sl(2) actions
sl(2) actions arise naturally on certain categories of D-modules and coherent sheaves. These actions can be used to construct equivalences of categories and subsequently knot invariants. After recalling the construction of these actions I will explain how they are related via the associated graded functor from D-modules to coherent sheaves.
Joel Kamnitzer, Categorical Lie algebra actions and braid group actions
I will discuss the notion of categorical Lie algebra actions, as introduced by Rouquier and Khovanov-Lauda. In particular, we will give examples of categorical Lie algebra actions on derived categories of coherent sheaves. I will show that such categorical Lie algebra actions lead to actions of braid groups. This gives many interesting examples of braid group actions on derived categories of coherent sheaves on resolutions of surface singularities, on resolutions of Slodowy slices, and, more generally, on quiver varieties.
Vassily Manturov (Internet talk), Parity in Low-Dimensional topology
Around 2004, Turaev defined a drastic simplification of virtual knots, which we call free knots: they are equivalence classes of Gauss diagrams without any decorations modulo Reidemeister moves; he conjectured these knots to be all trivial. We disprove this conjecture, and, moreover, prove that the cobordism classes of free knots are highly nontrivial. The main objective of our talk is the notion of parity. Parity can be axiomatized and used for some other purposes: for constructing functorial mapping from knots to knots, for proving minimality theorem, for improving lots of well known invariant. The question of existence of a parity for classical knots remains an open problem.
Igor Nikonov (Internet talk), Khovanov homology of graph links
Graph-links, introduced by D. Ilyutko and V. Manturov, are combinatorial analog of classical and virtual links. "Diagrams" of graph-links are simple undirected labeled graphs and graph-links themself are equivalence classes of the graphs modulo formal Reidemeister moves. In knot theory graph-links appear as intersection graphs of rotating chord diagrams of links. It is known that intersection graph determines chord diagram uniquely up to mutations. Thus any link invariant that does not distinguish mutant links is a candidate for an invariant of graph-links. For example, Alexander polynomial, Jones polynomial and HOMFLY are of such type. The problem is the equivalence relation of the graph-links can be too strong so there can exist two chord diagrams which have different values of the invariant but whose intersection graphs coincide as graph-links. So the problem of finding a natural construction that would extend this the invariant from intersection graphs to graph-links is nontrivial. For Jones polynomial this problem was solved by D. Ilyutko and V. Manturov.
Matthew Hedden, Recent results in concordance
Under an equivalence relation called concordance, knots form a group with operation provided by the connected sum. This group depends on the category (smooth or topological) in which it is defined, and highlights interesting aspects of four-dimensional topology. My talk will focus on attempts to understand the group through its subgroups; that is, given a collection of knots one can consider the subgroup which they generate within the larger concordance group. I will discuss various geometrically relevant subgroups and some recent progress. This is joint work with subsets of {Paul Kirk, Chuck Livingston, Danny Ruberman}.
Lev Rozansky, Categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in S2 x S1
This is a joint work with M. Khovanov. The WRT invariant of a link L in S2 x S1 at high level can be expressed as an evaluation of a special polynomial invariant of L at prime root of unity. We categorify this polynomial invariant by associating to L a bi-graded homology whose graded Euler characteristic is equal to this polynomial.
If L is presented as a circular closure of a tangle t in S2 x S1, then the homology of L is defined as the Hochschild homology of H_n-bimodule associated to t by Khovanov in one of his old papers. This homology can also be expressed as a stable limit of Khovanov homology of the circular closure of t in S3 through a torus braid with high twist.