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Khovanov homology and restricted isotopies
by
Benjamin Audoux
Université Paul Sabatier, Toulouse, France
Coauthors: Thomas Fiedler
Usually, knots are given by diagrams and isotopies by sequences of Reidemeister moves. According to the orientation, one can distinguish two Reidemeister moves of type II and eight of type III. By forbiding some of them, we define some restricted notion of isotopies of links, that is braid-like and star-like isotopies.
Jones polynomial and Khovanov homology can be refined in order to be invariant only under these isotopies. Studying them may shed new light on internal mechanism of usual Khovanov homology.
Paper reference: arXiv:math.GT/0503080
Date received: November 20, 2005
Torsion in the chromatic cohomology of graphs
by
Michael Chmutov
The Ohio State University
This is a work in progress aimed to prove that the chromatic cohomology of graphs has only Z2 torsion. I will explain the idea of the proof and demonstrate it on two examples of complete graphs K4 and K5.
Date received: November 22, 2005
Extreme parts of the Khovanov complex
by
Sergei Chmutov
The Ohio State University, Mansfield
There are natural upper and lower bounds for the degree (second grading) in the Khovanov complex of a link. We call the corresponding subcomplexes the extreme bottom and extreme top subcomplexes respectively. I will talk about a description of these complexes in terms of independence vertex complexes of some graphs related to the link diagram. The result should be regarded as a categorification of the Bae-Morton formulas for the extreme coefficients of the Jones polynomial.
Date received: November 22, 2005
A Khovanov-type cohomology theory for graphs
by
Laure Helme-Guizon
GWU
Coauthors: Y. Rong and J. Przytycki, GWU
In recent years, there have been a great deal of interests in Khovanov homology theory For each link L, Khovanov defines a family of homology groups whose "graded" Euler characteristic is the Jones polynomial of L. These groups were constructed through a categorification process which starts with a state sum of the Jones polynomial, constructs a group for each term in the summation, and then defines boundary maps between these groups appropriately for each positive integer n.
It is natural to ask if similar categorifications can be done for other invariants with state sums.
In the first part of this talk, we establish a homology theory that categorifies the chromatic polynomial for graphs. We show our homology theory satisfies a long exact sequence which can be considered as a categorification for the well-known deletion-contraction rule for the chromatic polynomial. This exact sequence helps us to compute the homology groups of several classes of graphs. In particular, we point out that torsions do occur in the homology for some graphs. This is joint work with Yongwu Rong, George Washington University, Washington DC, USA
The second part of this talk will discuss for which graphs these homology groups have torsion and provide some computational results. This is joint work with Yongwu Rong and Józef Przytycki, George Washington University, Washington DC, USA
Paper references:
[1] A categorification for the chromatic polynomial, LHG, Yongwu Rong : arXiv:math.CO/0412264 or http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-53.abs.html
[2] Graph Cohomologies from Arbitrary Algebras, LHG, Yongwu Rong arXiv:math.QA/0506023
[3] LHG, Y. Rong and J. Przytycki, Torsion in Graph Homology, arXiv:math.GT/0507245
Paper reference: arXiv:math.CO/0412264, arXiv:math.QA/0506023, arXiv:math.GT/0507245
Date received: December 6, 2005
An application of TQFT: determining the girth of a knot
by
Lisa Hernandez
University of California, Riverside
Coauthors: Xiao-Song Lin
A knot diagram can be divided by a circle into two parts, such that each part can be coded by a planar tree with integer weights on its edges. A half of the number of intersection points of this circle with the knot diagram is called the girth. The girth of a knot is then the minimal girth of all diagrams of this knot. The girth of a knot minus 1 is an upper bound of the Heegaard genus of the 2-fold branched covering of that knot. We will use TQFTs coming from the Kauffman bracket to determine the girth of some knots. Consequently, our method can be used to determine the Heegaard genus of the 2-fold branched covering of some knots.
Date received: November 11, 2005
A categorification for the Tutte polynomial
by
E. Fanny Jasso-Hernandez
George Washington University
Coauthors: Yongwu Rong
In 2004, inspired by M. Khovanov's graded homology theory, L. Helme-Guizon and Y. Rong constructed a graded homology theory for graphs. The graded Euler characteristic of these cohomology groups yields the chromatic polynomial.
Using similar ideas, given a graph G, we explain how to define a chain complex in such a way that the graded Euler Characteristic of its cohomology groups is essentially the Tutte polynomial. This is joint work with Yongwu Rong.
Date received: December 6, 2005
Non-sharpness of the Morton Franks Williams inequality
by
Keiko Kawamuro
Columbia University
I will exhibit an infinite sequence of fibered 4-braids on which the MFW inequality is not sharp. For this, I use the enhanced Milnor numbers of the fiber surfaces.
Paper reference: arXiv:math.GT/0509169
Date received: November 16, 2005
Triply-graded link homology and Hochschild homology of Soergel bimodules
by
Mikhail Khovanov
Columbia University
There exists a triply-graded link homology theory with the HOMFLY-PT polynomial as the Euler characteristic. I'll describe this theory using the language of Hochschild homology and Soergel bimodules.
Paper reference: arXiv:math.GT/0510265
Date received: November 28, 2005
What is categorification?
by
Mikhail Khovanov
Columbia University
Categorification turns numbers into vector spaces and vector spaces into categories. This talk will be a broad overview of categorification, together with examples of the latter and applications.
Date received: December 4, 2005
Knots with the same Jones polynomial
by
Kerry Luse
GWU
Coauthors: Yongwu Rong
This talk is motivated by the open question of whether or not the Jones polynomial can detect knottedness. We show that given a knot of the form K = K̅m# Km, one can construct K¢ such that K and K¢ are distinct, but indistinguishable by the Jones polynomial. This example is motivated by the work of Eliahou, Kauffman, and Thistlethwaite [2001], Kanenobu [1986], and Watson [2005].
Date received: December 8, 2005
The Knot Atlas
by
Scott Morrison
UC Berkeley
Coauthors: Dror Bar-Natan
I'll give a quick tour of the new Knot Atlas, a publically editable website (a wiki), which is both an atlas of knots (for humans) and a database of invariants (for computers).
Date received: November 29, 2005
Burnside Kei (involutory quandle)
by
Maciej Niebrzydowski
The George Washington University
Coauthors: Jozef H. Przytycki
Burnside Kei is an involutory quandle, Q, satisfying the universal relation a = ... *a*b* ... *a*b, for any a, b in Q. The simplest nontrivial relation of this sort is a=b*a*b, which is equivalent to commutativity, a*b=b*a. To every link we can associate its n-th Burnside Kei that is invariant under n-moves. In this joint work with Jozef Przytycki we investigate some algebraic properties of such structures.
Date received: December 5, 2005
On torsion in the first A3 graph cohomology
by
Milena D. Pabiniak
George Washington University
Coauthors: Jozef H. Przytycki, Radmila Sazdanovic
We will show that for a simple graph G the first
cohomology HA31, 2v(G)−3 (G) behave "nicely" under
one-vertex product (star product), that is:
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Additionally we conjecture that every simple graph G containing K4, complete graph of 4 vertices, will also contain a Z2 torsion in HA31, 2v(G)−3(G) cohomology (we have computed that HA31, 5(K4) = Z33 ÅZ2 ÅZ2 ).
Date received: December 4, 2005
Confluence of Khovanov homology and Hochschild homology: application to truncated polynomial algebra
by
Jozef H. Przytycki
George Washington University
Half a year ago (precisely May 4, 2005) I noted that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2, n)-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of Khovanov-Rozansky, sl(n), homology and their deformations we conjecture the connection. The best framework to explore our ideas is to use a comultiplication-free version of Khovanov homology for graphs developed by L. Helme-Guizon and Y. Rong. In this framework we prove that for any commutative unital algebra A and A-bimodule M the Hochschild homology of A with coefficients in M is isomorphic to M-reduced graph homology over A of a polygon. In this talk we show how one can extend the results from a polygon to some other graphs. This part is mostly speculative, and we concentrate on truncated polynomial algebra Am = Z[x]/(xm). We conjecture, in particular, that if a simple graph G contains a triangle than HAm1, (m−1)(v−1) − (m−2)(G) contains Zm. The last part of the talk describes joint work with U.Kaiser, M. Pabiniak, R. Sazdanovic and A. Shumakovitch.
Paper reference: arXiv:math.GT/0402402, arXiv:math.GT/0507245, arXiv:math.GT/0509334
Date received: December 5, 2005
Integral Bases for Certain TQFT-Modules of the Torus
by
Khaled Qazaqzeh
Louisiana State University
Gilmer and Masbaum, extending the work with van Wamelen in genus one, found explicitly bases for naturally defined lattices over a ring of algebraic integers in the SO(3)-TQFT-modules of surfaces at roots of unity of odd prime order. We find similar bases for the SU(2)-TQFT- modules of the torus.
Paper reference: arXiv:math.GT/0509565
Date received: November 30, 2005
A quadruply-graded graph homology for the Bollobas-Riordan polynomial
by
Yongwu Rong
The George Washington University
Following M. Khovanov, there have been a number of homology theories for graphs which categorify various graph polynomials. They include
My apology for any possible omissions.
This talk focuses on the Bollobas-Riordan polynomial for ribbon graphs. This is a three variable polynomial which contains all the polynomials involved above. We construct a quadruply-graded homology theory which categories the Bollobas-Riordan polynomial.
Date received: December 8, 2005
A categorification of the 2-variable and SU(N) HOMFLY-PT polynomials side by side
by
Lev Rozansky
University of North Carolina at Chapel Hill
Coauthors: M. Khovanov
This is a report on our joint work with M. Khovanov. I will explain how a 2-variable HOMFLY-PT polynomial is categorified with the help of simple Koszul complexes and how this categorification is modified for the SU(N) case by introduction of an extra differential which turns Koszul complexes into Koszul matrix factorizations.
Date received: November 18, 2005
On the properties of the first graph cohomology over the algebra of truncated polynomials Am
by
Radmila Sazdanovic
George Washington University
Coauthors: Jozef Przytycki,
Milena Pabiniak
We created Mathematica package for computing HA31, (v−1)(m−1)−(m−2)(n−1)/2(G) and Tor HAm1, (v−1)(m−1)−(m−2)(n−1)/2(G) for an arbitrary simple graph with v vertices, any n ³ 3 and algebras of truncated polynomials Am. Obtained results motivated following conjectures:
Conjecture 1. For all simple graphs G and G1: G Ì G1 Þ HA31, 2v(G)−3(G) Ì HA31, 2v(G1)−3(G1).
We verified this for the complete lattice of all non-isomorphic subgraphs of K5.
Conjecture 2. For any complete graph with n vertices Kn, n > 3 the following is true: H1, 2n−3A3(Kn) = Z3n−1 ÅZ2 ÅZn(n−1)(2n−7)/6.
We verified this for n £ 20 and for K20 indeed we get: H1, 37A3 = Z319 ÅZ2 ÅZ2090.
Moreover, we will present a few interesting observations:
1. Computations suggest that if K4 Ì G then Z2 Ì HA31, 2v−3 (G). Similarly, if K5 Ì G then Z2 ÅZ34 Ì HA31, 2v−3(G).
2. The only example (so far) of Z4 torsion appears in Tor(HA51, 14(K6)) = Z540 ÅZ226 ÅZ4.
Date received: December 6, 2005
Khovanov homology is stronger than the colored Jones polynomial
by
Alexander Shumakovitch
GWU
There are knots which can be distinguished by their Khovanov homology but which have the same the colored Jones polynomial.
Date received: December 21, 2005
Unoriented TQFTs and link homology
by
Paul Turner
Institut de Recherche Mathématique Avancée, Strasbourg, France
Coauthors: V. Turaev
I will discuss the notion of unoriented topological quantum field theory and provide a classification result in dimension 1+1 in terms of Frobenius algebras with additional structure. A restrictive class of these can be applied to Bar-Natan's geometric setting to obtain link homologies for virtual links. This is joint work with V. Turaev.
Paper reference: arXiv:math.GT/0506229
Date received: November 24, 2005
Hopf diagrams
by
Alexis Virelizier
University of California, Berkeley
Coauthors: Alain Bruguieres
I will describe a universal encoding of string links in terms of Hopf algebraic structures. For this, I will introduce the notion of Hopf diagrams. Computing a quantum invariant of a 3-manifold reduces then to the purely formal computation of the associated Hopf diagram, followed by the evaluation of this diagram in a given category.
Paper reference: arXiv:math.QA/0505119
Date received: November 17, 2005
Khovanov-Rozansky homology and a graphical calculus for tensor products
by
Ben Webster
UC Berkeley
We describe a more flexible definition of Khovanov-Rozansky homology, relating knot diagrams and modules by means of a graphical calculus. As time allows, we will show how this perspective allows for easy proofs of Reidemeister invariance, the skein relation on Euler characteristics, and how it leads us to computer computations (and a hope for more efficient ones).
Date received: November 29, 2005
Copyright © 2005 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas.