Invited Speaker:

Paul Turner
Title: Khovanov homology of torus knots T(3,N)

Abstract: This will be a relatively elementary talk in which I will explain how to calculate Khovanov homology for the family of torus knots T(3,N). I will explain how to set up a spectral sequence which organises in a convenient way the iterated use of the usual skein long exact sequence and then apply this to torus knots.











Contributed Speaker:

John Armstrong
Title: Functors extending the Kauffman bracket

Abstract: The Kauffman bracket is a family of ring-valued regular isotopy invariants of knots and links defined by a “skein relation”. We want to define functors on apropriate categories of tangles that recover the bracket invariants when restricted to knots and links. We proceed by recognizing the skein-theoretic approach to knot theory as a type of representation theory. This gives rise to a category of functors deeply related to categories of bilinear forms, which may give insights into the structure of Bracket categorifications like Khovanov homology.












Contributed Speaker:

Benjamin Audoux

Title : Heegaard-Floer homology for singular links

Abstract : Using the combinatorial description for knot Heegaard Floer homology, we give a generalization to singular knots which does fit in the general program of categorification of Vassiliev finite type invariants theory.










Contributed Speaker:

Christian Blanchet
Title: On functoriality of sl(N) link homology

Abstract: We have used a trivalent TQFT to define an homology of links which should be equivalent to Khovanov-Rozansky sl(N) link homology. We obtain a Lee-Rasmussen type spectral sequence converging to a degenerate theory, and establish strong functoriality.

This will be an extended version of my talk at Liverpool, available on Christian Blanchet web page.









Contributed Speaker:

Paolo Ghiggini
Title: Knot Floer homology, contact structures, and fibred knot.

Abstract: In this talk I will describe a strategy to prove that knot Floer homology detects fibred knots using taut foliations and contact structures. This strategy has been carried out by me for genus-one knots in S^3, and by Yi Ni in the general case.

for the genus-one case: http://lanl.arxiv.org/abs/math.GT/0603445
for the general case: http://lanl.arxiv.org/abs/math.GT/0607156










Contributed Speaker:

João Martins
Title: The Fundamental Crossed Module of the Complement of a Knotted Surface

Abstract: We define an algorithm for calculating the fundamental crossed module Pi_2(M,M^1) where M is the complement of a knotted surface in S^4 and M^1 is the 1-skeleton of a handle decomposition of it. Both the handle decomposition and the algorithm are defined from a hyperbolic spliting of M. This in particular yields a completely geometric method for calculating the algebraic 2-type of the complement of a knotted surface.










Contributed Speaker:

Gregor Masbaum
Title : Integral structures in TQFT

Abstract : The pretext for proposing this talk at this conference is the problem of categorifying the Reshetikhin-Turaev invariant of closed 3-manifolds. As pointed out by Khovanov, here it might be useful to exploit the fact that in the appropriate normalization, the 3-manifold invariants are not just arbitrary complex numbers, but algebraic integers, which might be easier to categorify.
However I do not know how to perform this categorification. Instead, my talk would be about my joint work with Pat Gilmer on what the integrality means for the TQFT-representations of mapping class groups associated with the quantum invariant. We get representations by matrices with explicitly computable integral coefficients (and one can wonder about the categorification of those ...)










Contributed Speaker:

Scott Morrison
Title: Functoriality for Khovanov homology in S^3.

Abstract: I’ll explain why Khovanov homology is functorial for links in S^3, not just B^3 as previously known.











Contributed Speaker:

Hendryk Pfeiffer
Title: Where is the braided monoidal 2-category?

Abstract: The extension of Khovanov homology to link cobordisms satisfies the Carter-Rieger-Saito version of the Roseman moves (in its original version only up to a sign though). These moves are part of the defining relations of a braided monoidal 2-category. To this day, the only known non-trivial example of such a 2-category seems to be the free one: 2-tangles. I explain how to get further examples from Khovanov homology.

Joint work with Aaron Lauda. The categorical part is work in progress, for the topological part see math.GT/0606331.











Contributed Speaker:

Marko Stosic
Title: sl(N)-link homology using foams and the Kapustin-Li formula - II

Abstract: In joint work with Marco Mackaay and Pedro Vaz, we define an almost topological construction of a rational link homology categorifying the sl(N)-link invariant. This construction uses foams which generalize the ones introduced by Khovanov in [1]. The evaluation of closed foams uses the Kapustin-Li formula, adapted to the context of foams by Khovanov and Rozansky [2]. We conjecture that our link homology theory is equivalent to Khovanov and Rozansky's in [3]
In this talk I will show how to use the Kapustin-Li formula in order to evaluate the closed foams.

Main references:
[1] M. Khovanov, sl(3) link homology, Alg.Geom.Top. 4(2004), 1045-1081.
[2] M. Khovanov and L. Rozansky, Topological Landau-Ginzburg models on a world-sheet foam, hep-th/0404189.
[3] M. Khovanov and L. Rozansky, Matrix factorizations and link homology, QA/0401268.














Contributed Speaker:

Pedro Vaz
Title: sl(N)-link homology using foams and the Kapustin-Li formula - I

Abstract: In joint work with Marco Mackaay and Marko Stosic, we define an almost topological construction of a rational link homology categorifying the sl(N)-link invariant. This construction uses foams which generalize the ones introduced by Khovanov in [1]. The evaluation of closed foams uses the Kapustin-Li formula, adapted to the context of foams by Khovanov and Rozansky [2]. We conjecture that our link homology theory is equivalent to Khovanov and Rozansky's in [3]
In this talk I will concentrate on the topological aspects of this theory.

Main references:
[1] M. Khovanov, sl(3) link homology, Alg.Geom.Top. 4(2004), 1045-1081.
[2] M. Khovanov and L. Rozansky, Topological Landau-Ginzburg models on a world-sheet foam, hep-th/0404189.
[3] M. Khovanov and L. Rozansky, Matrix factorizations and link homology, QA/0401268.














Contributed Speaker:

Vladimir Vershinin
Title: Yamada Polynomial and Khovanov Cohomology

Abstract: For any graph we define bigraded cohomology groups whose graded Euler characteristic is a multiple of the Yamada polynomial of the graph.











Contributed Speaker:

Benjamin Webster
Title: A geometric model of the Hochschild homology of Soergel bimodules. (joint w/ Geordie Williamson)

Abstract: Since the calculation of the Hochschild homology of Soergel bimodules is a key step in the calculation of Khovanov and Rozansky’s triply graded homology theory, this homology is an object of considerable interest. We will show how this homology can be identified with the equivariant intersection cohomology of certain subvarieties of SL(n,C). When this subvariety is smooth, this allows us to give an explicit presentation of the Hochschild homology.