Topology Atlas | Conferences


Knots in Washington XVIII; Khovanov homology
May 28-30, 2004
George Washington University
Washington, DC, USA

Organizers
Jozef H. Przytycki (GWU), Yongwu Rong (GWU)

Conference Homepage


Abstracts

Generalizations of Khovanov homology: links on surfaces and beyond.
by
Marta Asaeda
University of Iowa
Coauthors: Jozef H. Przytycki, Adam S. Sikora

I will talk about some definitions of Khovanov homologies on surfaces, and miscellaneous homologies of other stuff like tangles constructed by similar idea.

Paper reference: arXiv:math.QA/0403527

Date received: May 25, 2004


4Tu
by
Dror Bar-Natan
University of Toronto

I will describe the 4Tu relation and tell you how it seems to encompass pretty much everything known about sl_2 Khovanov homology: the original theory, tangles and cobordisms, the Lee variant, the Asaeda-Przytycki-Sikora variant and a new variant which works only over Z/2.

Of course, the newly released Khovanov-Rozansky homology is more exciting. But I don't understand it in the least.

Date received: May 24, 2004


Quandle invariants and constructing cocycles
by
Scott Carter
University of South Alabama
Coauthors: Masahico Saito, Shin Satoh

In this talk, we outline a variety of contexts in which quandle cocycle invariants can be applied. We indicate new constructions of non-trivial cocycles.

Paper reference: arXiv:math.GT/0401183

Date received: April 19, 2004


A-polynomial and Bloch invariants of hyperbolic 3-manifolds
by
Abhijit Champanerkar
Barnard College, Columbia University

For an one-cusped hyperbolic 3-manifold N with an ideal triangulation we construct a plane curve in C^2 using the combinatorics of the triangulation. We show that the defining polynomial of this curve is the PSL(2, C) A-polynomial of N. The Bloch invariant of N is defined using the triangulation of N and determines the volume of N. We relate the A-polynomial to the variation of Bloch invariant of N.

Date received: May 27, 2004


The Kauffman bracket and the Bollobas-Riordan polynomial of ribbon graph
by
Sergei Chmutov
The Ohio State University, Mansfield
Coauthors: Igor Pak

For a ribbon graph G we consider an alternating link LG in the 3-manifold G×I represented as the product of the oriented surface G and the unit interval I. We show that the Kauffman bracket [LG] is an evaluation of the recently introduced Bollobas-Riordan polynomial RG. This results generalizes the celebrated relation between Kauffman bracket and Tutte polynomial of planar graphs. Joint work with Igor Pak.

Paper reference: arXiv:math.GT/0404475

Date received: April 30, 2004


Burnside groups in knot theory
by
Mieczyslaw K. Dabkowski
University of Texas at Dallas
Coauthors: Jozef H. Przytycki (GWU)

In classical Knot Theory a substantial amount of effort has been devoted to the search for unknotting moves on links. Tangle moves which unknot or at least simplify significantly every link play an important role in Knot Theory and 3-dimensional topology. Good understanding of these moves allows us to analyze skein modules obtained by their deformations. For example, a deformation of a crossing change leads to the Jones polynomial of links and more generally to the Homflypt skein module relation. We answer the question whether rational [p/q]-moves (p an odd prime) can simplify any link into a trivial one. We show that pth Burnside group of a link can be an obstruction to the simplification. In particular we disprove Harikae-Nakanishi (2, 2)-move conjecture, showing that the knots 940 and 949 cannot be trivialized. We also answers Kawauchi's question, showing that half" 2-cabling of the Whitehead link cannot be reduced into a trivial ink via 4-moves.

Paper reference: arXiv:math.GT/0309140

Date received: May 27, 2004


On the coefficients of the Jones polynomial
by
Oliver Dasbach
Louisiana State University
Coauthors: Xiao-Song Lin

We show that certain combinations of coefficients of the Jones polynomial of alternating knots give linear upper and lower bounds for the hyperbolic volume of the knot complement. We compare it with the non-alternating case.

Paper reference: arXiv:math.GT/0403448

Date received: May 25, 2004


An analog of Khovanov homology for Graphs
by
Laure Helme-Guizon
GWU
Coauthors: Yongwu Rong

This talk will explain how, from a graph, one can construct a chain complex whose graded Euler characteristic is the Chromatic polynomial of the graph we started from.

We will also describe an exact sequence which relates the homology groups of G, G-e and G/e, for any graph G and any edge e of G.

Date received: May 17, 2004


Categorifications of quantum group representations.
by
Mikhail Khovanov
University of California, Davis

We give examples of categories whose Grothendieck groups are various representations of simple Lie algebras and quantum groups, and explain how they relate to known and conjectural link homology theories.

Date received: May 18, 2004


Matrix factorizations and link homology I
by
Mikhail Khovanov
University of California, Davis
Coauthors: Lev Rozansky

For each positive n we construct a bigraded homology theory of links whose Euler characteristic is the quantum sl(n) invariant.

Paper reference: arXiv:math.QA/0401268

Date received: May 16, 2004


sl(3) link homology
by
Mikhail Khovanov
University of California, Davis

We introduce the notion of sl(3) foam and show how it leads to a categorification of the quantum sl(3) link polynomial.

Paper reference: arXiv:math.QA/0304375

Date received: May 26, 2004


Spanning trees and Khovanov homology
by
Ilya Kofman
Columbia University
Coauthors: Abhijit Champanerkar, Oleg Viro

Thistlethwaite showed that the Jones polynomial is a state sum over spanning trees of the Tait graph, obtained by checkerboard coloring a knot diagram. We show that there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. In fact, the spanning tree complex is a deformation retract of Khovanov's complex. For alternating knots, this complex is the simplest possible because all boundary maps are zero.

This is work in progress, joint with Abhijit Champanerkar and Oleg Viro.

Date received: May 25, 2004


Kauffman Skein Module of the Projective Space
by
Maciej Mroczkowski
Uppsala Univ.

The Kauffman skein module of the projective space is computed. It is free and generated by an infinite set of links. This set may be chosen to be {L_n, n in N or n=0}, where L_n is an arbitrary link consisting of n projective lines for n>0, and L_0 is an affine unknot.

Paper reference: arXiv:math.GT/0312205

Date received: May 25, 2004


Thickness of Khovanov homology of k-almost alternating links
by
Jozef H. Przytycki
George Washington University
Coauthors: Marta M. Asaeda (University of Iowa)

We give a size restriction on the Khovanov homology of almost alternating links. We prove, in particular, that if D is a non-split k-almost alternating diagram without a nugatory crossing, then D is H-(k, k)-thick and TH-(k, k-1)-thick.

Paper reference: arXiv:math.GT/0402402

Date received: May 27, 2004


Experimenting with Exact Sequences
by
Yongwu Rong
George Washington University
Coauthors: Laure Helme-Guizon

We demonstrate some simple minded computations using the exact sequences in Khovanov type homologies, both for graphs that we defined and for the framed knots introduced by Viro. This is mostly joint work with Laure Helme-Guizon.

Date received: May 27, 2004


Matrix factorizations and link homology II
by
Lev Rozansky
University of North Carolina at Chapel Hill
Coauthors: Mikhail Khovanov

We will review the properties of matrix factorizations and their relation to graphs, tangles and foams.

Paper reference: arXiv:hep-th/0404189

Date received: May 16, 2004


Torsion of the Khovanov homology
by
Alexander Shumakovitch
Dartmouth College

This talk is (obviously) devoted to the torsion of the Khovanov homology groups. Although the Betti numbers of these groups have many remarkable properties, their torsion appear to be even more fascinating. We are going to prove several properties of this torsion, discuss methods of its calculation and finally formulate several conjectures about it.

Paper reference: arXiv:math.GT/0405474

Date received: May 27, 2004


Khovanov Homology and Skein Modules
by
Adam S. Sikora
SUNY Buffalo
Coauthors: Marta Asaeda, Jozef H. Przytycki

Kauffman bracket invariant leads to the Khovanov homology groups of links and to skein modules of 3-manifolds. We will describe an interesting relation between these two important objects.

Date received: May 27, 2004


The completeness of the involutory Kuperberg invariant
by
Fernando Souza
University of Iowa

The involutory Kuperberg invariant of oriented 3-manifolds, in its most general form (categorical, diagrammatic), is complete for oriented, closed 3-manifolds, providing an algebraicization of them as morphism of Hopf-algebra objects. After a review of the definition of the invariant, we discuss the reconstruction of 3-manifolds from any given value of the invariant, as well as the completeness of the invariant, revising the original approach.

Date received: May 27, 2004


What to categorify? Face models?
by
Oleg Viro
MSRI and Uppsala University

What to categorify? Face models?

Date received: May 28, 2004


Copyright © 2004 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.