


Knot Homologies
Org: Dror BarNatan (Toronto) [PDF]
 CARMEN CAPRAU, The University of Iowa, Iowa City, Iowa 52242
A sl(2) tangle homology for dotted, seamed cobordisms
[PDF] 
We construct a sl(2) BarNatan like tangle homology for dotted,
seamed cobordisms with Z[a] coefficients. This theory is
functorial under link cobordisms. A version of the original Khovanov
Homology corresponds to the choice a=0. Likewise, for a=1 we
recover Lee's theory.
 OLIVIER COLLIN, Université du Québec à Montréal
Nontrivial actions on Floer homology
[PDF] 
Given a finite order orientationpreserving diffeomorphism t: Y^{3} ® Y^{3} of an integer homology sphere Y^{3}, it is an
interesting and difficult general problem to understand the effect of
the induced map on instanton Floer homology. In this talk, we
consider oddorder diffeomorphisms of integer homology sphere and
their induced effect on SU(2)character varieties and Floer
homology. We are interested in cases where the resulting equivariant
Casson invariant differs from the Casson and the consequences for
finding nontrivial actions on Floer homology. This provides the
first examples of nontrivial oddorder actions on the Floer homology
of irreducible homology spheres. We shall also see how this can be
related to the existence of incompressible surfaces in the
3manifold Y^{3}.
 JOEL KAMNITZER, UC Berkeley, Berkeley, CA
Knot homology via derived categories of coherent sheaves
[PDF] 
We will give a construction of a knot homology theory using the
derived category of coherent sheaves on a certain variety arising in
geometric representation theory.
 MIKHAIL KHOVANOV, Columbia University
Braid cobordisms and triangulated categories
[PDF] 
We review known actions of the braid group on triangulated categories
and their extensions to representations of the category of braid
cobordisms.
 PETER LEE, University of Toronto, 40 St. George St., Toronto, Ontario,
Canada
ClosedForm Associators in a "Toy" Quotient
[PDF] 
In the theory of finitetype invariants of knots, a key objective is
to construct a "universal finitetype invariant", which takes values
in a certain target space of chord diagrams. Construction of such a
universal invariant can typically be reduced to determining its value
on a particular kind of tangle, which represents a sort of
"associativity" property. This value is usually referred to as an
"associator", and it is hoped that finding an explicit expression
for such an associator could offer significant knottheoretic
insights. In general, however, finding associators has proved to be
an extremely difficult problem.
In this talk I will discuss how a closedform associator can be
constructed in a particular "toy" quotient of the usual target space,
which may ultimately help us to construct associators in more general
target spaces.
 ROBERT LIPSHITZ, Columbia University, 2990 Broadway, New York, NY 10027, USA
Some recent developments in HeegaardFloer homology
[PDF] 
We will discuss some recent developments in HeegaardFloer
homologyprogress on making it combinatorial and attempts to
generalize it.
 GAD NAOT, University of Toronto, Canada
The Universal Khovanov Link Homology TheoryExtracting
Algebraic Information
[PDF] 
In this talk I will present the universal Khovanov link homology
theory (n=2). This theory is developed using the full strength of
the geometric formalism of Khovanov link homology theory and has many
computational and theoretical advantages. The universal theory
answers questions regarding the amount of algebraic information held
within the complex associated to a link. It also answers questions
regarding the extraction of this information by giving control over
the various TQFTs applied to the complex (along with control over
other gadgets such as the various spectral sequences related to these
TQFTs). After a brief overview and some reminders I will introduce
the major tools and ideas used in developing the universal theory
(such as surface classification, genus generating operators, complex
isomorphisms and "promotions"). Then, I will present some of the
advantages of such a theory, time permitting (more on the topic can be
found at arXiv:GT/0603347).
 JUAN ARIEL ORTIZNAVARRO, University of Iowa, Iowa City, IA
Khovanov Homology and Reidemeister Torsion
[PDF] 
The Reidemeister Torsion construction can be applied to the chain
complex that is used to compute the Khovanov homology of a knot or a
link. This defines a volume form on Khovanov homology. The volume
form transforms correctly under Reidemeister moves to give an
invariant volume on the Khovanov homology. We use this to study the
invariants of knots.
 JAKE RASMUSSEN, Princeton University, Princeton, NJ, USA
Stable KRhomology of torus knots
[PDF] 
Computer calculations suggest that the limiting behavior of the
Khovanov homology of T(m,n) as n ® ¥ is rather complicated.
In contrast, the corresponding limit for the HOMFLY homology of
T(m,n) is quite simple. I'll describe how to calculate this limit
and explain why the result provides evidence for the presence of a
symmetry in the HOMFLY homology.
 LEV ROZANSKY, University of North Carolina at Chapel Hill
Virtual knots, convolutions and a categorification of the
SO(2N) Kauffman polynomial
[PDF] 
We present a categorification construction for the SO(2N)
specialization of the Kauffman polynomial and prove its invariance
under the first and second Reidemeister moves. The construction
follows the KauffmanVogel alternating sign formula, which expresses
the Kauffman polynomial of a link in terms of polynomials of 4valent
planar graphs. We define the matrix factorization associated to the
4vertex as a convolution of a chain of two saddle morphisms, relating
parallel and virtually crossing pairs of arcs.
This is a joint work with M. Khovanov.
 PAUL SEIDEL, MIT, Room 2270, 77 Massachusetts Ave., Cambridge, MA 02139
Localization in Floer homology and applications
[PDF] 
I will explain how to construct localization maps for Z/2actions in
Floer theory, and how this explains the relation between the
symplectic version of Khovanov homology and OzsvathSzabo theory.
This is joint work with Ivan Smith.
 DYLAN THURSTON, Columbia

 ROBB TODD, University of Iowa
Khovanov Homology and the Twist Number of Alternatinng
Knots
[PDF] 
O. Dasbach and X. S. Lin showed that the sum of the absolute value of
the second and penultimate coefficient of the Jones polynomial of an
alternating knot is equal to the twist number of the knot. Here we
give a new proof of their result using a variant of Khovanov's
homology that was defined by O. Viro for the Kauffman bracket. The
proof is by induction on the number of crossings using the long exact
sequence in Khovanov homology corresponding to the Kauffman bracket
skein relation.

