LowDimensional Topology and Categorification
Videos
All talks will be in Room S240, in the basement of the math department building.

 Mikhail Khovanov,
Diagrammatics for categorified quantum groups
 Categorified quantum groups are monoidal categories
and 2categories 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 4genus of knots
 The nonorientable 4genus of a knot is the minimum first betti number of a nonorientable surface in the 4ball 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 CassonGordon theory, to obstructions arising via HeegaardFloer theory
 Jacob Rasmussen,
Handle decompositions and exotic S^{2}xS^{2}'s
 Akhmedov and Park have recently constucted an exotic S^{2}xS^{2}. 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 4handles.
 Liam Watson,
Lspaces and leftorderability (No Video Available)
 Examples suggest that there is a correspondence between Lspaces and 3manifolds with fundamental group that cannot be leftordered. This talk will introduce these notions, present certain families of examples, and discuss some related questions.
 Aaron Lauda,
Extended graphical calculus for categorified sl_{2}
 Igor Frenkel conjectured that the quantum enveloping algebra of sl_{2} 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 2category that is the idempotent completion, or Karoubi envelope, of a 2category 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 sl_{2} (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 KhovanovLauda 2category
 Robert Lipshitz,
Introduction to bordered Heegaard Floer homology
 Bordered Floer homology is an extension of OzsvathSzabo's invariant HFhat to 3manifolds 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 HFhat of closed 3manifolds and, time permitting, the maps induced by 4dimensional 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 3manifolds 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 secondextremal 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 ndimensional 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 ncategory.
 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 KhovanovSeidel quiver algebras and sutured Khovanov homology
 OzsvathSzabo's spectral sequence from Khovanov homology to Heegaard Floer homology has generated a number of interesting applications to questions in lowdimensional topology. By combining the constructions of OzsvathSzabo 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 HeegaardFloer 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 KhovanovSeidel. Along the way, we will discuss an intriguing relationship between these KhovanovSeidel bimodules and certain bimodules appearing in the bordered Floer package of LipshitzOzsvathThurston.
 Sabin Cautis,
Dmodules and categorical sl(2) actions
 sl(2) actions arise naturally on certain categories of Dmodules 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 Dmodules 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 KhovanovLauda. 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 LowDimensional 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
 Graphlinks, introduced by D. Ilyutko and V. Manturov, are combinatorial analog of classical and
virtual links. "Diagrams" of graphlinks are simple undirected
labeled graphs and graphlinks themself are equivalence classes of
the graphs modulo formal Reidemeister moves. In knot theory
graphlinks 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 graphlinks. For example, Alexander polynomial, Jones
polynomial and HOMFLY are of such type. The problem is the
equivalence relation of the graphlinks can be too strong so there
can exist two chord diagrams which have different values of the
invariant but whose intersection graphs coincide as graphlinks. So
the problem of finding a natural construction that would extend this
the invariant from intersection graphs to graphlinks 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 fourdimensional 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)
WittenReshetikhinTuraev invariant of links in
S^{2} x S^{1}
 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 bigraded 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_nbimodule
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.