Topology Atlas | Conferences

Knots in Washington XXIV; Dedicated to the memory of Xiao-Song Lin
April 13-15, 2007
George Washington University
Washington, DC, USA

Jozef H. Przytycki (GWU and UMD), Yongwu Rong (GWU), Alexander Shumakovitch (GWU)

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Vassiliev invriants that do not distinguish mutant knots.
Sergei Chmutov
The Ohio State University, Mansfield
Coauthors: Sergei Lando

The purpose of this presentation is to describe all Vassiliev invariants that do not distinguish mutant knots in terms of their weight systems. Namely, a (canonical) Vassiliev invariant does not distinguish mutant knots if and only if its weight system depends on the intersection graph of a chord diagram only.

Date received: March 26, 2007

Modeling protein-DNA complexes using tangles.
Isabel Darcy
University of Iowa

Protein-DNA complexes have been modeled using tangles. A tangle consists of arcs properly embedded in a 3-dimensional ball. The protein is modeled by the 3D ball while the segments of DNA bound by the protein can be thought of as arcs embedded within the protein ball. This is a very simple model of protein-DNA binding, but from this simple model, much information can be gained. The main idea is that when modeling protein-DNA reactions, one would like to know how to draw the DNA. For example, are there any crossings trapped by the protein complex? How do the DNA strands exit the complex? Is there significant bending? Tangle analysis cannot determine the exact geometry of the protein-bound DNA, but it can determine the overall entanglement of this DNA, after which other techniques may be used to more precisely determine the geometry.

Date received: March 26, 2007

Skein modules and their ramifications
Uwe Kaiser
Boise State University

I will give a survey of the theory of skein modules and skein algebras focusing on:

1. historical origins, and the idea of algebraic topology based on knots,

2. problems of calculation, general principles and viewpoints, and sample results,

3. the role of skein modules and algebras within quantum topology (representation theory, TQFT)

Date received: March 30, 2007

Bar-Natan algebras and modules of oriented surfaces
Uwe Kaiser
Boise State University

Asaeda and Frohman initiated the study of Bar-Natan skein modules, which are modules defined from surfaces embedded in 3-manifolds. The (relative) Bar-Natan skein modules of F ×I, for F an oriented surface, can be considered as the modules of a tautological TQFT for the geometric Khovanov homology on surfaces. In this case the Bar-Natan modules are naturally modules over Bar-Natan algebras. I will discuss some recent results and ideas about these modules based on mapping class group action and incompressible surface theory in F ×I. These results should be related with the work of Asaeda, Przytycki and Sikora on Khovanov homology for oriented surfaces.

Date received: March 30, 2007

Formal knot theory and clock homology
Louis H Kauffman

We discuss formal knot theory and clock homology

Date received: April 21, 2007

Links, Milnor's invariants, and robust manifolds
Slava Krushkal
University of Virginia

Milnor's invariants provide an obstruction for a link to be slice, and more generally to be homotopically trivial. I will discuss an extension of this theory to links bounding disjoint more general, "robust", 4-manifolds in the 4-ball. This problem is motivated by the 4-dimensional topological surgery conjecture.

Date received: April 5, 2007

p-adic framed braids and p-adic framed links
Sofia Lambropoulou
National Technical University of Athens
Coauthors: Jesus Juyumaya

We first present the construction of the p-adic framed braids. These can be viewed as classical framed braids, but with the framings being p-adic integers, or as natural infinite cablings of modular framed braids. We then construct the p-adic Yokonuma-Hecke algebras. The classical Y-H algebras can be defined as quadratic quotients of the modular framed braid group algebras. Finally, we construct a p-adic Markov trace on the p-adic Y-H algebras, which we normalize to obtain an isotopy invariant of p-adic framed links.

Date received: April 14, 2007

Q 3-Stranded Quantum Algorithm for the Jones Polynomial
Samuel J. Lomonaco, Jr.
University of Maryland Baltimore County (UMBC), Baltimore, MD 21250
Coauthors: Louis H. Kauffman and Samuel J. Lomonaco, Jr.

Let K be a 3-stranded knot (or link), and let L denote the number of crossings of K. Let ε₁ and ε₂ be two positive real numbers such that ε₂≤1.

We create two algorithms for computing the value of the Jones polynomial V_{K}(t) at all points t=exp(iϕ) of the unit circle in the complex plane such that |ϕ|≤2π/3.

The first algorithm, called the classical 3-stranded braid (3-SB) algorithm, is a classical deterministic algorithm that has time complexity O(L). The second, called the quantum 3-SB algorithm, is a quantum algorithm that computes an estimate of V_{K}(exp(iϕ)) within a precision of ε₁ with a probability of success bounded below by 1-ε₂. The execution time complexity of this algorithm is O(nL), where n is the ceiling function of (ln(4/ε₂))/2ε₁². The compilation time complexity, i.e., an asymptotic measure of the amount of time to assemble the hardware that executes the algorithm, is O(L).

Date received: April 9, 2007

Knots with identical Khovanov Homology, after Watson
Kerry Luse
Coauthors: Y. Rong

In a paper with Y. Rong, we showed that there is an infinite family of distinct knots with the same Jones polynomial. A recent result by L. Watson gives examples of knots with the same Khovanov homology. He uses the classification given in our paper to show that there is an infinite family of distinct knots with identical Khovanov homology. I will outline Watson's construction and the main ideas in his proof.

Date received: April 12, 2007

Hochschild homology and 3-manifolds
Michael McLendon
Washington College

Given a Heegaard splitting of a closed 3-manifold, the zeroth Hochschild homology of the skein algebra of the gluing surface with coefficients in the tensor product of the skein modules of the two handlebodies can be seen as the Kauffman bracket skein module of the resulting 3-manifold. It is natural to ask if the higher Hochschild homology modules are independent of the Heegaard splitting and thus also invariants of the 3-manifold. We will look at the Hochschild chain complex associated to a Heegaard splitting and investigate the effect of stabilization.

Date received: March 25, 2007

A polynomial invariant of finite quandles
Sam Nelson
Whittier College

Unlike groups, quandles have no single identity element; rather, the trivial action is distributed throughout the set. We will see how this distribution can be quantified for finite quandles as a two-variable polynomial which completely determines the quandle structure for quandles of order up to 4, and as an application we will see a new family of refinements of the quandle counting invariants.

Date received: March 22, 2007

Burnside groups of knots, tangle moves and their skein module deformations
Jozef H. Przytycki
Coauthors: Mieczyslaw K. Dabkowski (UT Dallas)

We describe the use of Burnside groups in knot theory and topology of 3-dimensional manifolds. We define the nth Burnside group, Bn(G), of the given group G, as the quotient of G by a normal subgroup generated by all elements of G of the form wn. If G is a free group of k generates we obtain the classical (1902) Burnside group B(k, n). The nth Burnside group of a link L in S3 is defined to be Bn(L) = Bn1(ML(2))), where ML(2) is the double branch cover of S3 branched along L. Our method of Burnside group of links allows us to settle (disprove) conjectures of Montesinos-Nakanishi, Kawauchi, and Harikae-Nakanishi-Uchida about 3-moves, 4-moves, and (2, 2)-moves, respectively. We can also show that the manifold Mφ (suspected of being the smallest volume hyperbolic oriented 3-manifold) is not obtainable from (S1 ×S2)k by a finite sequence of q/5-surgeries.

Skein modules, on which most of research was concentrated till now, are based on the deformation of a smoothing (1-move) and crossing change (2-move). We argue that deformation of tangle moves (e.g. 3-moves, 4-moves, (2, 2)-moves) can lead to new, powerful skein modules of 3-manifolds.

Date received: April 11, 2007

Feynman diagrams and transition polynomial
Yongwu Rong
George Washington University
Coauthors: Kerry Luse

Feynman diagrams arise from physics, and have interesting applications in mathematics and molecular biology. The transition polynomial for 4-regular graphs was defined by Jaeger to unify polynomials given by vertex reconfigurations similar to the skein relations of knots. It is closely related to the Kauffman bracket, Tutte polynomial, and the Penrose polynomial. This talk will discuss some natural relations between the two concepts. In particular, we show that the genus of a Feynman diagram is encoded in the transition polynomial. This is joint work with Kerry Luse.

Date received: April 12, 2007

Cohomology for Frobenius algebras and applications
Masahico Saito
University of South Florida

We investigate possible cohomology theories for Frobenius algebras from two approaches. The first is from point of view of deformations of the compatibility condition of Frobinius algebras between multiplication and comultiplication, following the graph calculus of cohomology for self-distributive morphisms developed earlier for coalgebras and Hopf algebras. The second is an analogue of the categorifications of the bracket and chromatic polynomial, based on 4-valent spatial graphs and smoothings. Possible applications are discussed for knot and manifold invariants and in relation to DNA recombinations of ciliates.

Date received: April 6, 2007

Unknotting number and unlinking gap
Radmila Sazdanovic
The George Washington University
Coauthors: Slavik Jablan

Computing unlinking number is a very difficult and complex problem. Therefore we define BJ-unlinking number and BJ-unlinking gap which will be computable due to the algorithmic nature of their definition. According to Bernhard-Jablan conjecture unknotting/unlinking number is the same as the BJ- unlinking number. We compute BJ-unlinking number for various families of knots and links for which the unlinking number is unknown and give experimental results for families of rational links with arbitrarily large BJ-unlinking gap and polyhedral links with constant non-trivial BJ-unlinking gap.

Date received: April 13, 2007

Khovanov homology for concatenation of tangles
Adam S Sikora
SUNY Buffalo

Khovanov defined homology theory for links and later generalized it to tangles. Other versions of tangle homology were proposed by Bar-Natan and Asaeda-Przytycki-Sikora. Unfortunately, in all of these homology theories, homology of tangles T_1 and T_2 seem not to determine the homologies of the link T_1 cup T_2 obtained by full concatenation of T_1 and T_2.

We propose a stronger version of homology theory for tangles which solves this problem - we prove a Kunneth-type formula for our homology, expressing homology of (T_1 cup T_2) by homologies of T_1 and of T_2.

Date received: April 6, 2007

Mutation invariance of Khovanov homology over Z/2Z
Stephan Wehrli
Columbia University

Using ideas of Dror Bar-Natan, we will show that Khovanov homology with coefficients in Z/2Z is invariant under component-preserving link mutation.

Date received: April 9, 2007

The Khovanov-Rozansky cohomology and Bennequin inequalities
Hao Wu
University of Massachusetts, Amherst

I will talk about some new developments in Khovanov-Rozansky cohomology, and explain how to use these to re-prove and generalize several Bennequin type inequalities.

Date received: April 4, 2007

Copyright © 2007 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.