Topology Atlas | Conferences

Abstracts

The Extension of "Topological-Style" Link Invariants to Tangles
by
John Armstrong
Yale University

Classical link invariants can be roughly divided into two classes - those which arise from combinatorial analyses of link diagrams and those which arise from the topology of the link embedding. This shows up in two distinct techniques for extending invariants to functors on categories of tangles.

This talk discusses the "topological-style" invariants, including the link group and the quandle of a link. The construction realizes the 2-category of tangles in a 2-category of cospans, as introduced by Bénabou, and seems to generalize to many other invariants of this type.

Date received: April 11, 2006

Unified Witten-Reshetikhin-Turaev invariant for integral and rational homology 3-spheres
by
Anna Beliakova
University of Zurich
Coauthors: Thang Le

In 2001, Habiro announced a construction of an invariant of integral homology 3-spheres with values in the universal cyclotomic ring. Habiro's invariant specializes at a root of unity to the value of the sl(2) Witten-Reshetikhin-Turaev invariant at that root. In this talk, we will explain how such a unified invariant can be consrtucted for integral and rational homology 3-spheres. Furthermore, we show that this unified invariant separates integral homology Seifert fibered spaces.

Date received: April 17, 2006

Introduction to Quandles and Their Cohomology
by
J. Scott Carter
University of South Alabama
Coauthors: Alissa Crans, Mohamed Elhamdadi, Matias Graña, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford, Masahico Saito, Shin Satoh

In this talk, I present the Joyce, Matveev, representation of quandles as inner automorphisms on a coset space. This representation is related to the fundamental quandle as a knot classifier. I will present some other examples of quandles and describe the basic idea of quandle cohomology. I will finish the talk with a discussion of

algebras, co-algebras, shelf algebras, and their related diagrammatic descriptions. This will provide a seque to Masahico Saito's talk later during the conference.

Date received: April 17, 2006

Quandles and colagebras
by
Alissa Crans
Ohio State University
Coauthors: Scott Carter, Mohamed Elhamdadi, and Masahico Saito

After examining how to obtain a solution to the Yang-Baxter equation and quandle structure from a Lie algebra, we discuss how to formulate similar structures in other algebraic settings. In particular, we will focus on quandles coming from coalgebras and Hopf algebras, and understand when such structures provide a solution of the Yang-Baxter equation.

Date received: May 10, 2006

Biquandles and knots invariants
by
Mohamed Elhamdadi
University of South Florida
Coauthors: Scott Carter and Masahico Saito

Biquandles are related to set theoretic solutions of the Yang-Baxter equation. Most examples known come from generalizations of the Burau representation.

We will give examples coming from Wada's representations of the braid groups as free group automorphisms and use them to study virtual knots.

Date received: May 1, 2006

The Turaev-Viro and the related TQFT in the case of Group Category
by
Petit Jerome
UM2, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5

We study the invariant of Turaev-Viro constructed by Gelfand and Kazhdan (this is nearly the same approach of Barrett-Westburry) with a special spherical category : Group category. In fact it is a spherical category whose all simple objects are inversible. Under other asumptions on this category we recover the invariant of Dijkgraaf-Witten. Moreover in this case the TQFT is more explicit.

Date received: April 11, 2006

Quandles with good involutions, linear biquandles and knot invariants
by
Seiichi Kamada
Hiroshima University

Quandle homology groups induce invariants, called quandle cocycle invariants, of knots or surface knots in 4-space. For calculation of the invariants, it is essential that knots or surface knots are oriented. On the other hand, the knot quandle can be generalized to the case where knots or surface knots are not oriented. Here we introduce the notion of a quandle with good involution, and a quandle cocycle invariant. We can use them for the study of unoriented knots and surface knots. If there is time, I would also like to introduce a virtual knot invariant, derived from a certain kind of linear biquandle, on which Roger Fenn, Naoko Kamada and I are working.

Date received: April 9, 2006

A categorification of the Burau representation
by
Mikhail Khovanov
Columbia University

I'll review a toy example of categorification, that of the Burau representation of the braid group. The talk is based on an old joint paper with Paul Seidel.

Date received: April 28, 2006

Symmetries in the SL(3, C)-Character Variety of a Rank 2 Free Group
by
Sean Lawton
University of Maryland

Denote the free group on two letters by F2 and the SL(3, C)-representation variety of F2 by R=Hom(F2, SL(3, C)). There is a SL(3, C)-action on the coordinate ring of R, and the geometric points of the subring of invariants is an affine variety X. It is has been shown by A. Sikora that X corresponds to the SU(3)-skein module of a 3-manifold with fundamental group F2. We determine explicit minimal generators for the subring of invariants which exhibit Out(F2) symmetries and allow for a succinct expression of the defining relations.

Date received: April 27, 2006

Disoriented and confused: fixing the functoriality of Khovanov homology
by
Scott Morrison
UC Berkeley
Coauthors: Kevin Walker

I’ll describe a modification of Bar-Natan’s cobordism model for Khovanov homology, introducing “disorientations”, and some rules for manipulating them. Using these, we discover that link cobordisms now induce honest well-defined maps between the complexes associated to links, not just up-to-sign maps.

In addition to “disorientations”, we can also add “confusions”: places where a disorientation changes type. These live up to their name; they need a spin framing, and change sign when rotated. However, they fix a few defects of the “disoriented” model, allow nice proofs, and make contact with some familiar features of su(2)’s planar representation theory.

There will be lots of pictures!

Date received: May 3, 2006

Quandle difference invariants
by
Sam Nelson
University of California, Riverside
Coauthors: Natasha Harrell

We will describe a method of detecting non-classicality of virtual knots using the fact that a virtual knot may have non-isomorphic upper and lower quandles. This method can be used with any invariant of virtual knots.

Date received: April 25, 2006

On tangle embeddings; quandles and quandle polynomials
by
Maciej Niebrzydowski
The George Washington University

I'll discuss recent work on applications of quandles to embeddings of tangles into long knots. We describe invariants obtained from quandle colorings and quandle longitudes that can be used to find obstructions to tangle embeddings.

Date received: May 5, 2006

On the first group of A₃ cohomology of graphs
by
Milena Pabiniak
George Washington University
Coauthors: Jozef H. Przytycki, Radmila Sazdanovic

We are analyzing properties of the first group of A₃ graph cohomology. We concentrate on the grading implied by the interpretation of Hochschild homology as graph homology of a polygon, that is (1, 2v-3).
We work with homology because this approach enables us to establish straightforward connections to the homology of simplicial complexes built on our graph. Precisely, the group H_{0, 2v-3}^A₃(G) can be almost entirely recovered from homology of two cell complexes with graph G as 1-skeleton:

• X_{3, 4}(G) obtained from G by adding 2-cells along 3-cycles and 4-cycles of G,

• X_{(3), 4}(G) obtained from G by adding 2-cells along 4-cycles of G and by identifying edges of every 3-cycle in G in a coherent way.

Date received: May 3, 2006

Spin Networks and SL(2, C)-Character Varieties
by
Elisha Peterson
University of Maryland, College Park
Coauthors: Sean Lawton

Denote the free group on 2 letters by F_2 and the SL(2,C)-representation variety of F_2 by R=Hom(F_2,SL(2,C)). The group SL(2,C) acts on R by conjugation. We construct an isomorphism between the coordinate ring C[SL(2,C)] and the ring of matrix coefficients, providing an additive basis of C[R]^SL(2,C) in terms of spin networks. Using a graphical calculus, we determine the symmetries and multiplicative structure of this basis. This gives a canonical description of the regular functions on the SL(2,C)-character variety of F_2 and a new proof of a classical result of Fricke, Klein and Vogt.

Date received: April 27, 2006

Combinatorial patterns in Khovanov type graph homology motivated by Hochschild homology
by
Jozef H. Przytycki
George Washington University
Coauthors: Milena Pabiniak (GWU) and Radmila Sazdanovic (GWU)

The algebra of truncated polynomials A_m = Z[x]/(x^m) plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. It is not difficult to compute Hochschild homology of A_m and the only torsion, Z_m, appears in grading (i, [(m(i+1))/2]) for any odd i. We have demonstrated that Hochschild homology is closely related to Khovanov homology via comultiplication free graph homology. We analyze here grading of graph cohomology which is producing torsion for a polygon. We find completely the cohomology H^{1, v-1}_A2(G) and H^{1, 2v-3}_A3(G). In the following talks the case H^{1, 2v-3}_A3(G) is described by my coauthors. Here we compute the first cohomology of a graph with underlying algebra A₂. We notice that working with the dual chain complex is much easier to visualize. In particular, we show that H₀^A₂(G) is a "symmetric" homology of a graph, that is the boundary of an edge is equal the sum of its endpoints.

Date received: May 5, 2006

Some differentials on Khovanov-Rozansky homology
by
Jacob Rasmussen
Princeton

I'll discuss recent progess on some conjectures (joint w/ N. Dunfield and S. Gukov) regarding the structure of Khovanov and Rozansky's sl(N) and HOMFLYPT homologies.

Date received: April 26, 2006

An isomorphism between Tutte homology for graphs and Khovanov homology for links
by
Yongwu Rong
George Washington University

The Tutte homology, defined by the speaker and E. Fanny Jasso-Hernandez, is a Khovanov type homology corresponding to the Tutte polynomial. It is well-known that the Tutte polynomial of a plane graph evaluated on the curve xy = 1 is essentially the Jones polynomial of the corresponding alternating link. In this talks, we show a categorification of this identity, namely, an isomorphism between these homology groups. This work is very preliminary.

Date received: May 4, 2006

Some new developments in quandle cohomology theory - computations, applications and quandles in tensor categories
by
Masahico Saito
University of South Florida
Coauthors: Too Many To List

Continuing Scott Carter's talk, more recent developments in the theory of quandle cohomology and its applications will be discussed. In particular, an overview will be given on computations of quandle cocycle invariants with polynomial cocycles, and their data base presented in our web site. An application in tangle embeddings is presented as a demonstration on how the data base can be used for applications of quandle cocycle invariants to topological properties of knots. Then structures similar to quandles in coalgebras and Hopf algebras will be discussed. For these structures, coboundary operators are constructed in low dimensions, towards their cohomology theories. This is an analogue of the Hochschild double complex for bialgebras. Deformation theoretical aspects and diagrammatics for cocycles will be discussed for both Hochschild cohomology and these quandle coboundary operators.

Date received: April 25, 2006

Torsion in the first group of the chromatic graph cohomology over algebras Z[x]/(x^m)
by
Radmila Sazdanovic
George Washington University
Coauthors: Jozef H. Przytycki, Milena Pabiniak

In this talk we focus on the first chromatic graph cohomology over algebra Z[x]/(x^m) of truncated polynomials. In particular, we are interested in grading motivated by the interpretation of Hochschild homology as graph cohomology of polygons and its generalization to arbitrary graphs. As an introduction to this talk, my coauthors will give the complete description of H^{1, v-1}_A2(G) and H^{1, 2v-3}_A3(G). However, for Z[x]/(x^m) and m > 3 we give only conjectures based both on theory developed for m=2, 3 and computational results.

Theorem 1.
For complete graph with n vertices K_n, n ≥ 4 have

H1, 2n-3A3(Kn) = Z2⊕Z3n-1⊕Z[(n(n-1)(2n-7))/6].

Corollary 2.
If a graph G contains a triangle then H^{1, 2v(G)-3}_A3(G) contains Z₃ torsion.

We show that (∀n) (∃ simple G) Z_n ∈  torH^{1, 2v-3}_A3(G).

Conjecture 3.
For any graph W_n with n vertices where one vertex is of degree n-1 and all the rest are of degree 3 (wheel), n > 4 and m ≥ 4 the following holds:

H1, 4n-3Am(Wnout)=Zmn-1 ⊕Zn-2

H1, 4n-3Am(Wn)=Zmn ⊕Zn

H1, 4n-3Am(Wnin)=Zmn-2⊕Zn-2

Conjecture 4.
For complete graph K_n where n ≥ 4 and odd m > 3 the following is true:

H1, (m-1)(n-2)+1Am(Kn) = Zm[(n (n-1)(n-2))/6] ⊕Z2[(n (n-1)(n-2)(n-3))/24] ⊕Z[(n (n-1)(n-2)(n-3))/12]

Moreover, we will present some computational results for width of H¹_A3(G) of several families of graphs graphs.

Date received: May 4, 2006

Turaev torsion of 3-manifolds with boundary
by
Christopher Truman
University of Maryland

I will talk on results relating leading order terms of Turaev torsion to certain "determinants" in cohomology (which can arise from cup products, or more generally Massey products) for 3-manifolds with boundary. These are analogous to some results of Turaev for closed 3-manifolds.

Date received: May 10, 2006

Copyright © 2006 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.