
Organizers 
Vassiliev invriants that do not distinguish mutant knots.
by
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 proteinDNA complexes using tangles.
by
Isabel Darcy
University of Iowa
ProteinDNA complexes have been modeled using tangles. A tangle consists of arcs properly embedded in a 3dimensional 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 proteinDNA binding, but from this simple model, much information can be gained. The main idea is that when modeling proteinDNA 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 proteinbound 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
by
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
BarNatan algebras and modules of oriented surfaces
by
Uwe Kaiser
Boise State University
Asaeda and Frohman initiated the study of BarNatan skein modules, which are modules defined from surfaces embedded in 3manifolds. The (relative) BarNatan 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 BarNatan modules are naturally modules over BarNatan 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
by
Louis H Kauffman
UIC
We discuss formal knot theory and clock homology
Date received: April 21, 2007
Links, Milnor's invariants, and robust manifolds
by
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", 4manifolds in the 4ball. This problem is motivated by the 4dimensional topological surgery conjecture.
Date received: April 5, 2007
padic framed braids and padic framed links
by
Sofia Lambropoulou
National Technical University of Athens
Coauthors: Jesus Juyumaya
We first present the construction of the padic framed braids. These can be viewed as classical framed braids, but with the framings being padic integers, or as natural infinite cablings of modular framed braids. We then construct the padic YokonumaHecke algebras. The classical YH algebras can be defined as quadratic quotients of the modular framed braid group algebras. Finally, we construct a padic Markov trace on the padic YH algebras, which we normalize to obtain an isotopy invariant of padic framed links.
Date received: April 14, 2007
Q 3Stranded Quantum Algorithm for the Jones Polynomial
by
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 3stranded 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 3stranded braid (3SB) algorithm, is a classical deterministic algorithm that has time complexity O(L). The second, called the quantum 3SB 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 1ε₂. 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
by
Kerry Luse
GWU
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 3manifolds
by
Michael McLendon
Washington College
Given a Heegaard splitting of a closed 3manifold, 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 3manifold. It is natural to ask if the higher Hochschild homology modules are independent of the Heegaard splitting and thus also invariants of the 3manifold. 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
by
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 twovariable 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
by
Jozef H. Przytycki
GWU and UMD
Coauthors: Mieczyslaw K. Dabkowski (UT Dallas)
We describe the use of Burnside groups in knot theory and topology of 3dimensional manifolds. We define the nth Burnside group, B_{n}(G), of the given group G, as the quotient of G by a normal subgroup generated by all elements of G of the form w^{n}. 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 S^{3} is defined to be B_{n}(L) = B_{n}(π_{1}(M_{L}^{(2)})), where M_{L}^{(2)} is the double branch cover of S^{3} branched along L. Our method of Burnside group of links allows us to settle (disprove) conjectures of MontesinosNakanishi, Kawauchi, and HarikaeNakanishiUchida about 3moves, 4moves, and (2, 2)moves, respectively. We can also show that the manifold M_{φ} (suspected of being the smallest volume hyperbolic oriented 3manifold) is not obtainable from (S^{1} ×S^{2})^{k} by a finite sequence of q/5surgeries.
Skein modules, on which most of research was concentrated till now,
are based on the deformation of a smoothing (1move) and crossing change
(2move). We argue that deformation of tangle moves (e.g. 3moves, 4moves,
(2, 2)moves) can lead to new, powerful skein modules of 3manifolds.
References:
http://arxiv.org/abs/math.GT/0501539
http://front.math.ucdavis.edu/math.GT/0205040
http://front.math.ucdavis.edu/math.GT/0309140
http://front.math.ucdavis.edu/math.GT/0405248
http://front.math.ucdavis.edu/math.GT/0109029
http://front.math.ucdavis.edu/math.GT/0010282
http://www.maths.warwick.ac.uk/gt/GTMon4/paper21.abs.html
Date received: April 11, 2007
Feynman diagrams and transition polynomial
by
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 4regular 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
by
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 selfdistributive morphisms developed earlier for coalgebras and Hopf algebras. The second is an analogue of the categorifications of the bracket and chromatic polynomial, based on 4valent 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
by
Radmila Sazdanovic
The George Washington University
Coauthors: Slavik Jablan
Computing unlinking number is a very difficult and complex problem. Therefore we define BJunlinking number and BJunlinking gap which will be computable due to the algorithmic nature of their definition. According to BernhardJablan conjecture unknotting/unlinking number is the same as the BJ unlinking number. We compute BJunlinking 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 BJunlinking gap and polyhedral links with constant nontrivial BJunlinking gap.
Date received: April 13, 2007
Khovanov homology for concatenation of tangles
by
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 BarNatan and AsaedaPrzytyckiSikora. 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 Kunnethtype 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
by
Stephan Wehrli
Columbia University
Using ideas of Dror BarNatan, we will show that Khovanov homology with coefficients in Z/2Z is invariant under componentpreserving link mutation.
Date received: April 9, 2007
The KhovanovRozansky cohomology and Bennequin inequalities
by
Hao Wu
University of Massachusetts, Amherst
I will talk about some new developments in KhovanovRozansky cohomology, and explain how to use these to reprove 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.