Topology Atlas | Conferences

Knots in Washington XXIII; Quandles, their homology and ramifications
November 17-19, 2006
George Washington University
Washington, DC, USA

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

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Polynomial Quandle Cocycles, Their knot invariants and Applications
Kheira Ameur

In this talk, a variety of n-cocycles for n ≥ 2 are constructed for some Alexander quandles, given by polynomial expressions. As an application, these cocycles are used to compute the invariants for (2, n)-torus knots, twist knots and their r-twist spins. The calculations in the case of (2, n)-torus knots resulted in formulas that involved the derivative of the Alexander polynomial. Non-triviality of some quandle homology groups is also proved using these cocycles. Another application is given for tangle embeddings. The quandle cocycle invariants are used as obstructions to embedding tangles in links. The formulas for the cocycle invariants of tangles are obtained using polynomial cocycles, and by comparing the invariant values, information is obtained on which tangles do not embed in which knots. Tangles and knots in the tables are examined, and concrete examples are listed.

Date received: November 13, 2006

Functors extending the Kauffman Bracket
John Armstrong
Yale University

We consider the problem of defining a monoidal functor on the category of framed tangles whose restriction to links is an evaluation of the Kauffman bracket. We show that a very large class of such functors correspond to bilinear forms with certain assymetry properties. This gives us an insight into why categorifications of the Bracket work they way they do.

Date received: October 30, 2006

A sl(2) tangle homology for seamed cobordisms with polynomial coefficients.
Carmen Caprau
The University of Iowa

We construct a sl(2) Bar-Natan like tangle homology for dotted, seamed cobordisms with Z[a] coefficients. This theory is functorial under link cobordisms. In the case of knots and links, a version of the original Khovanov Homology corresponds to the choice a = 0. Likewise, for a = -1 we recover Lee's theory.

Date received: October 31, 2006

Infiltration of elaborate schemes
Scott Carter
University of South Alabama
Coauthors: Alissa Crans, Mohamed Elhamdadi, Pedro Lopes, Masahico Saito

The common thread to the Jacobi identity, the associative rule, Moufang loops, and the self-distributivity axiom is that an initial collection of elements is multiplied in various ways, and there is an identity among these possible products. Starting from the coboundary map for a function on a magma, we construct a second boundary in these various contexts such that compositions of these maps is zero. The context is quite general. In the case of Lie algebras and algebras, this method gives the ordinary cohomology theory. In the case of self-distributive structures, we have a cohomology that includes quandle cohomology.

Date received: November 13, 2006

Quandle Queries
Alissa S. Crans
Loyola Marymount University
Coauthors: J. Scott Carter, Mohamed Elhamdadi, Masahico Saito

As the kickoff to the weekend we begin with an overview of quandles, including a definition and a handful of examples. Roughly speaking, a quandle is a set equipped with two binary operations satisfying axioms that capture the essential properties of the operations of conjugation in a group and algebraically encode the three Reidemeister moves. In preparation for numerous talks this weekend we will briefly describe quandle cohomology, invariants, and applications. We will continue by exploring the relationship between quandles and Lie algebras, emphasizing the significant role played by the self-distributive operations that each of these structures possess. In particular, we will consider how quandles provide a conceptual explanation of the passage from a Lie group to its Lie algebra.

Date received: November 13, 2006

Interpretation of quandle invariants in terms of knot group representations
Michael Eisermann
Institut Fourier, University of Grenoble

The classical knot group and the more recent invention of knot quandles are closely related concepts, and so it is not surprising to expect relationships between the various invariants derived from them. It is less obvious, however, to make the transition explicit and to establish a precise dictionary between both points of view. This endeavour is nevertheless important for the mutual benefit of the two approaches, and indispensable if we wish to exploit classical results in the quandle framework.

In this talk I will present some results that establish such an explicit correspondence. In particular my aim is to interpret the fundamental class in the second homology group of the knot quandle, and to represent quandle homology state-sum invariants by knot group representations.

Date received: November 10, 2006

The adjoint of Hopf algebras and cohomology
Mohamed Elhamdadi
University of South Florida
Coauthors: S. Carter, A. Crans, and M. Saito

The conjugation on groups has a analogue for Hopf algebras. A cohomology theory for the adjoint of Hopf algebras is proposed. Explicit examples of calculations in low dimensional cases are proposed. This is a joint work (in progress) with S. Carter, A. Crans and M. Saito.

Date received: November 8, 2006

A categorification for the Tutte polynomial for matroids: From a matroid M, we produce a chain complex whose graded Euler Characteristic is the Tutte polynomial of M.
Laure Helme-Guizon
George Washington University

In recent years, there has been a great deal of interest 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.

It is natural to ask if similar categorifications can be done for other invariants with state sums.

In 2004, Yongwu Rong and I established a homology theory which categorifies the chromatic polynomial for graphs.

Then, Yongwu Rong and Fanny Jasso-Hernandez came up with a categorification for the Tutte polynomial for graphs. This works really differs from ours since the graded Euler characteristic is a two-variable polynomial.

Matroids are a generalization of graphs (and of vectors spaces) so it is natural to ask if a categorification can be obtained for the Tutte polynomial for matroids.

In this talk, I will remind you briefly of what a matroid is (if there is any popular demand/need for that) and then I will show you a categorification for the Tutte polynomial for matroids. It is more than a cosmetic upgrade of the categorification for the Tutte polynomial for graphs by Yongwu Rong and Fanny Jasso-Hernandez since their construction relies heavily on the notion of connected components of a graph which doesnÂ’t exist for matroids.

I wish to thank Fanny Jasso-Hernandez and Joe Bonin (both from The George Washington University) for helpful discussions.

Date received: November 12, 2006

A categorifcation for the Penrose polynomial
Kerry Luse
George Washington University
Coauthors: Yongwu Rong

Given a connected plane graph, we construct cohomology groups whose Euler characteristic is the Penrose polynomial of the corresponding medial graph, evaluated at an integer. This work is motivated by Khovanov's categorifcation of the Jones polynomial for knots, and the subsequent categorifcations by Helme-Guizon and Rong, and Jasso-Hernandez and Rong, of the chromatic and Tutte polynomials, respectively.

Date received: November 15, 2006

Homology of dihedral quandles I
Maciej Niebrzydowski
The George Washington University
Coauthors: Jozef H. Przytycki

We prove nontriviality of the fourth quandle homology of dihedral quandles Rk, for odd prime k, and we discuss torsion in such homology. We also show that the rack homology Hn(Rk) contains Zk for k odd prime and n>2. Of particular importance in these proofs is the operation HRn(Rp)→ HRn+1(Rp), that turns out to be a monomorphism on homology. We expect that many of the techniques that we use can be generalized to other Alexander quandles.

Date received: November 15, 2006

Homology of dihedral quandles II
Jozef H. Przytycki
George Washington University
Coauthors: Maciej Niebrzydowski (GWU)

We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of the dihedral quandle satisfies H3R(Rp) = Z ⊕Zp for p odd prime (Ohtsuki, Conjecture 5.12). In particular, the quandle homology H3Q(Rp) = Zp. We conjecture that for n > 1 the quandle homology satisfies: HnQ(Rp) = Zpfn where fn are "delayed" Fibonacci numbers, that is fn = fn-1 + fn-3 and f(1)=f(2)=0, f(3)=1. We propose the method of approaching the conjecture by constructing rack homology operations HRn(Rp) → HRn+1(Rp) and HRn(Rp) → HRn+2(Rp), and quandle homology operations HRn(Rp) → HRn+2(Rp) and HRn(Rp) → HRn+3(Rp). We conjecture, and partially prove (as outlined in the previous talk by Maciej Niebrzydowski), that the above operations are monomorphisms and the images (in appropriate dimensions) are disjoint. To approach the general conjecture about HnQ(Rp) = Zpfn we need one more quandle homology operation HQn(Rp) → HQn+4(Rp), construction of which is an open problem.

References: T. Ohtsuki, Problems on invariants of knots and 3-manifolds, Geometry and Topology Monographs, Volume 4, 2003, 455-465; e-print:

Date received: November 15, 2006

A quick trip through combinatorial knot Floer homology
Yongwu Rong
George Washington University

This is a short lesson based on a few recent papers following a preprint by Manolescu-Ozsvath-Sarkar where a purely combinatorial description for the knot Floer homology is provided. It is aimed at beginners and may not be appropriate for experts. Much of the talk is based on concrete examples with pictures of "tik-tak-toe" and "guitar chords."

Date received: November 17, 2006

Quaternions and Rotations
Domingo Ruiz
Universite de Paris Sud / University of Maryland

Quaternions are mainly known as Hamilton's extension of complex numbers to a four-fold real division algebra. The average student of mathematics has little or almost not any familiarity with this number field, even though many of their properties are very accessible. We attempt to display some of these to reveal how quaternions clarify some links between geometry and algebra.

Date received: November 10, 2006

Applications, constructions, and interpretations of quandle cocycles
Masahico Saito
University of South Florida

In this talk, first an overview will be given on the three aspects mentioned in the title, for quandle cocycles and cocycke knot invariants. Then some new results and approaches will be discussed, such as relations between cocycle invariants and minimal Fox colors of knot diagrams, and constructions of quandle cocycles from a groupoid cohomology theory.

Date received: November 7, 2006

Virtual automorphisms as mapping classes
Dragomir Saric
Queens College CUNY
Coauthors: Sylvain Bonnot and Robert Penner

We consider the group Vaut(π1 (S)) of virtual automorphisms of a punctured surface group π1(S). By a result of C. Odden, this group is the mapping class group of a solenoid. We use Whitehead moves and an appropriate combinatorial object (the triangulation complex) to give generators and a presentation of Vaut(π1 (S)). The talk will be elementary with some background on the mapping class group of a surface.

Date received: November 10, 2006

The sheet numbers of 2-knots with non-trivial fundamental quandles
Kobe University

An embedded 2-sphere in R4 is called a 2-knot, which is described by a diagram through a projection of R4 onto R3. Such a diagram is regarded as a disjoint union of compact connected surfaces divided by crossing information. The sheet number of a 2-knot is the minimal number of such surfaces for all possible diagrams of the 2-knot. We prove that if the fundamental quandle of a 2-knot is non-trivial (in particular, if the fundamental group of the complement of the 2-knot is non-trivial), then its sheet number is greater than or equal to four. This gives an alternative proof of our previous result that the sheet numbers of the 0- and 2-twist-spun trefoils are equal to four.

Date received: October 25, 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.