Topology Atlas | Conferences


Knots in Washington XVII, Conference on Knot Theory and its Ramifications
December 19-21, 2003
GWU
Washington, DC, USA

Organizers
Marta M.Asaeda (U.Iowa), Mietek K.Dabkowski (UTD), Jozef H.Przytycki(GWU), Yongwu Rong (GWU)

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Abstracts

Khovanov homology of links in I-bundles over surfaces
by
Marta M. Asaeda
University of Iowa
Coauthors: Jozef H. Przytycki, Adam S. Sikora

Khovanov homology is defined on links in S^3, constructed as a categorification of Jones polynomial: the graded Euler characteristics of Khovanov homology coincide with Jones polynomial. In this talk I explain the generalization of Khovanov homology on link diagrams on surfaces other than RP^2, and mention the relation with Skein modules.

Date received: December 12, 2003


Gram determinant for type B Temperley-Lieb algebra
by
Gefry Barad
GWU

We sketch the idea which should lead to a proof of Rodica Simion conjecture and its generalizations.

Date received: December 19, 2003


Mahler measure of the Jones polynomial, part I
by
Abhijit Champanerkar
Barnard College, Columbia University
Coauthors: Ilya Kofman

The Mahler measure of the Alexander polynomial (Silver, Williams) and A-polynomial (Boyd, Rodriguez-Villegas) has been related to the volume of the knot complement. We show that the Mahler measure of the Jones polynomial converges under twisting in any link diagram. In this respect, the Jones polynomial, like the Alexander polynomial, behaves like hyperbolic volume under Dehn surgery. For torus knots, we obtain the explicit limit from the HOMFLY polynomial. The proof combines the representation theory of braid groups with linear skein theory. For twisting on two or three strands, we use the spanning tree expansion of the Jones polynomial to provide explicit formulas which extend previously known results.

Date received: December 10, 2003


On certain integral tensor categories over cyclotomic rings and integral TQFTs
by
Qi Chen
Sunny at Buffalo

We construct some tensor categories over cyclotomic rings. TQFT's coming from these tensor categories will be discussed.

Date received: December 19, 2003


Fundamental group of the double branch cover of S3 along 2-bridge knots
by
Mieczyslaw K. Dabkowski
University of Texas at Dallas
Coauthors: Jozef H. Przytycki (GWU), Amir A. Togha (GWU)

The associated core group of a link diagram, \Pi(D(2) was introduced by R.Fenn and C.Rourke (following the core group of Joyce).

\Pi(D(2) is the group associated to the diagram D as follows: generators of GD correspond to arcs of the diagram. Any crossing vs yields the relation rs=yiyj-1yiyk-1 where yi corresponds to the overcrossing and yj, yk correspond to the undercrossings at vs.

The topological interpretation of GD was given by M.Wada: \Pi(D(2) is the free product of the fundamental group of the cyclic branched double cover of S3 with branching set L and the infinite cyclic group. That is: \Pi(D(2)=\pi1((ML)(2)) * Z.

We give a very simple proof of the Wada theorem using Wirtinger presentation of the fundamental group of a link. We show how to use our construction to find (well organized) presentations of fundamental groups of double branched cover along 2-bridge knots.

In the follow up talk by A.Thoga the construction will be use to show that some of our groups are not left-orderable.

Date received: December 10, 2003


Flat Lorentz 3-Manifolds
by
Bill Goldman
University of Maryland

In 1977, Milnor asked whether a free group of rank 2 can act properly by affine transformations on R}3. He suggested that one might start with a Fuchsian group G_0 discretely embedded in SO(2,1) and deform it by "adding translations" to obtain a group G of affine transformations. However he said, "it seems difficult to determine whether the group acts properly." In 1983, Margulis showed that this is indeed possible, and in 1990 Drumm constructed examples using fundamental polyhedra. Drumm showed that every noncocompact torsionfree discrete subgroup of SO(2,1) admits proper affine deformations. Margulis introduced a class function A:G → R which is a ``signed marked Lorentzian length spectrum.'' Furthermore he showed that unless A(g) is positive (respectively negative) for all nontrivial elements of G, then the affine deformation is nonproper. It was natural to conjecture that this necessary condition is sufficient. In this talk I will describe an extension of this invariant to geodesic currents (joint work with Labourie and Margulis), relating this conjecture to the ergodic theory of the geodesic flow of convex cocompact hyperbolic surfaces, and will survey recent results on proper affine actions on R^3.

Date received: December 10, 2003


An analog of Khovanov Homology for graphs
by
Laure Helme-Guizon
GWU

From a graph we produce a chain complex. The construction is based on the idea used by Khovanov to produce a chain complex from a link diagram.

Date received: December 15, 2003


The program "LinKnot"-its theoretical background and experimental mathematics results obtained
by
Slavik Jablan
The Mathematical Institute, Knez Mihailova 35, P.O.Box 367, 11001 Belgrade, Serbia&Montenegro
Coauthors: Radmila Sazdanovic

The program "LinKnot" can be very efficiently used in experimental mathematics, as a tool for computing data (knot and link invariants, polynomials, signature, unknotting and unlinking numbers, etc.) for very large families of knots and links (KLs). From them, new conjectures can be made.

We will present the results connected to Bernhard-Jablan Conjecture on unknotting and unlinking numbers, projection gap, amphichirality, and Alexander polynomials derived in general form for diffrerent families of KLs. For example, a projection gap in rational KLs occurs for the first time for the link 414 with n=9 crossings. For n=10 the only case is the famous knot 514; for n=11 we have the knot 4142, and the links 434, 614, 4142, 5132, 51113 with the same property; for n=12 there are five such knots: 714, 534, 4143, 6132, 61113; for n=13 there are 7 knots 4414, 6142, 41314, 51322, 231412, 511132, 513112 and 16 links 616, 634,814, 5152, 5332, 6133, 7132, 34132, 41422, 51115, 61123, 71113, 241312, 411142, 611122 and 4211113. All thet results can be extended to infinite families of KLs.

Date received: November 10, 2003


Superbridge index of knots
by
Gyo Taek Jin
Korea Advanced Institute of Science and Technology

I will describe upper and lower bounds of superbridge index by other invariants.

Date received: December 17, 2003


Turaev-Viro invariants of 3-manifolds and normal surfaces
by
Joanna Kania-Bartoszynska
National Science Foundation and Boise State University
Coauthors: Charles Frohman

The formula for the Turaev-Viro invariant of a 3-manifold depends on a complex parameter t. When t is not a root of unity, the formula becomes an infinite sum. We analyze convergence of this sum when t does not lie on the unit circle, in the presence of an efficient triangulation of the three-manifold. The terms of the sum can be indexed by surfaces lying in the three-manifold. The contribution of a surface is largest when the surface is normal and when its genus is the lowest. This is joint work with Charles Frohman, University of Iowa.

Date received: December 13, 2003


The Block Property of Integral Bases for TQFT's and its Consequences
by
Thomas Kerler
The Ohio State University

TQFT's defined over general rings allows us to map topological information of 3-manifolds into the ideal structure of the ring. In many examples, such as the rings of cyclotomic integers, integral bases for the TQFT's have a block property with respect to the sewing of surfaces which implies a number of important structural properties, such as finite typeness, cut-number estimates, and special filtrations of the mapping class groups. We will explain these relations and connections in greater generality, and provide some concrete exmaples.

Date received: December 9, 2003


Two Types of Amphichiral Links
by
Mark Kidwell
U.S. Naval Academy

We define component preserving amphichiral (CPA) links and component switching amphichiral (CSA) links. We prove that an unoriented two-component link with non-zero even linking number cannot be CPA, answering a question of Livingston.

Date received: November 12, 2003


Mahler measure of the Jones polynomial, part II
by
Ilya Kofman
Columbia University
Coauthors: Abhijit Champanerkar

The Mahler measure of the Alexander polynomial (Silver, Williams) and A-polynomial (Boyd, Rodriguez-Villegas) has been related to the volume of the knot complement. We show that the Mahler measure of the Jones polynomial converges under twisting in any link diagram. In this respect, the Jones polynomial, like the Alexander polynomial, behaves like hyperbolic volume under Dehn surgery. For torus knots, we obtain the explicit limit from the HOMFLY polynomial. The proof combines the representation theory of braid groups with linear skein theory. For twisting on two or three strands, we use the spanning tree expansion of the Jones polynomial to provide explicit formulas which extend previously known results.

Date received: December 10, 2003


My first and hundredth papers: from Zn-actions on surfaces to periodicity of links and Khovanov homology
by
Jozef H. Przytycki
George Washington University
Coauthors: R. Anstee, M.M. Asaeda, S. Betley, D. Bullock, M.K. Dabkowski, P. Gilmer, J. Hoste, M. Ishiwata, W. Jakobsche, F. Jaeger, V.F.R. Jones, J. Kania-Bartoszynska, M. Lozano, K. Murasugi, D. Repovs, D. Rolfsen, W.Rosicki, A.S. Sikora, M.Sokolov, K. Taniyama, A.A. Togha, T. Tsukamoto, P. Traczyk, A. Yasuhara, T. Zukowski, T. Januszkiewicz, J. Dymara, S. Lambropoulou, D. Silver, S. Williams

I published my first paper 25 years ago. It was part of my master degree thesis written in Warsaw University under supervision of Agnieszka Bojanowska. I was then almost 25 five years old.

Today I am 50 so I hope I will be forgiven that my talk is about my first and hundredth papers.

1. Some remarks on actions of Zn-groups on 3-manifolds, Bull. Ac. Pol. Scie. Ser. Math. Astr. Phys XXVI (7) 1978, 625 - 633.

100. Khovanov homology of links in I-bundles over surfaces (with M.M.Asaeda and A.S.Sikora), preprint, 2003.

Date received: December 13, 2003


A Kauffman bracket polynomial for legendrian links
by
Yongwu Rong
George Washington University

Legendrian link arise naturally from contact structures in R^3. Here we present an invariant of legendrian links analogous to the Kauffman bracket for the usual links in topological category.

Date received: December 11, 2003


Knot theory program LinKnot
by
Radmila Sazdanovic
The Faculty of Mathematics, University of Belgrade, Serbia and Montenegro
Coauthors: Slavik Jablan (The Mathematical Institute, Knez Mihailova 35, P.O.Box 367 11001 Belgrade, Serbia and Montenegro)

The Mathematica-based Windows knot theory program LinKnot is the extension of the program Knot2000 (K2K) written by M.Ochiai and N.Imafuji. LinKnot provides solutions and tools for problems in knot theory and supports working with links (not only with knots). For the first time, input for a computer program is not Dowker code or graphics, but human-comprehensive Conway notation of KLs represented as a Mathematica string. For all KLs there is no restriction on the number of crossings. The program also provides the complete data base of alternating KLs with at most 12 crossings, non-alternating KLs with at most 11 crossings and the data base of basic polyhedra with at most 20 crossings.

LinKnot provides tools for drawing KLs, calculating all polynomial invariants of KLs, working with braids, KLs reduction, etc. The most significant result is computing unknotting and unlinking numbers, calculated according to Bernhard-Jablan Conjecture. Moreover, for all alternating KLs one can compute minimum Dowker codes, find all non-isomorphic projections, work with the graphs of KLs, compute linking numbers, breaking and spliting numbers, signatures, and many other KL invariants. All this makes creating of one’s own experiments in knot theory possible, while at the same time profiting from the visualization capabilities and fast and easy obtained results.

Date received: November 10, 2003


Torsion of the Khovanov Homology
by
Alexander Shumakovitch
Dartmouth College

Given a diagram D of an oriented link L in the 3-sphere, one can assign to it a family of Abelian groups Hi, j(D) using a construction due to Mikhail Khovanov. These groups are defined as homology groups of an appropriate (graded) chain complex C(D), and their isomorphism classes depend on the isotopy class of L only. The graded Euler characteristic of C(D) is a version of the Jones polynomial of L.

Although the ranks of the Khovanov homology groups have many remarkable properties, their torsion appear to be even more fascinating. In this talk, we prove several properties of this torsion, discuss methods of its calculation and finally formulate several conjectures about it.

Date received: December 10, 2003


Computing the Khovanov Homology with KhoHo
by
Alexander Shumakovitch
Dartmouth College

Given a diagram D of an oriented link L in the 3-sphere, one can assign to it a family of Abelian groups Hi, j(D) using a construction due to Mikhail Khovanov. These groups are defined as homology groups of an appropriate (graded) chain complex C(D), and their isomorphism classes depend on the isotopy class of L only. The graded Euler characteristic of C(D) is a version of the Jones polynomial of L.

One of the most remarkable properties of the Khovanov homology is that their ranks are much smaller than the ones of the original chain complex. For example, the chain groups of the torus knot (2, 11) have ranks as high as 11'000, but the homology themselves are at most Z. This property makes it utterly complicated to compute the Khovanov homology for knots and links with even 10 crossings.

In this talk, we show how one can dramatically reduce the size of the complex, without changing its homology, with the help of two elementary operations. This simplification procedure was implemented in the program KhoHo. We will see how to use KhoHo to compute and study Khovanov Homology.

Date received: December 10, 2003


Quantum Invariants of knots and links
by
Adam S. Sikora
SUNY Buffalo and IAS

We will provide an introduction to quantum invariants of knots - a topic which has been in the center of mathematical research for the last 20 years. We will describe the construction of quantum invariants using representation theory of quantum groups. (No prior knowledge of quantum groups is assumed). Finally, we will discuss an elementary (skein) approach to the SU_n-quantum invariants due to Kauffman and Kuperberg for n=2 and 3, and our generalization of their work for all n.

Date received: November 12, 2003


Skein theory for SU_n-quantum invariants
by
Adam S. Sikora
SUNY Buffalo and IAS

The Kauffman bracket skein relations provide an important method of studying SU_2-quantum invariants of links and 3-manifolds. Among its many applications, it makes possible to relate the SU_2-quantum invariants to Khovanov homology, skein modules, the (noncommutative) A-polynomial, and the SL_2-character varieties. In this talk, we discuss our skein calculus for SU_n-quantum invariants for all n, which hopefully has equally broad applications. Finally, we show that the SU_n-skein module of a 3-manifold M based on our skein relations is a q-deformation of the coordinate ring of the SL_n-character variety of pi_1(M).

Date received: November 12, 2003


Spherical Tetrahedra and a Semiclassical Invariant of Three-Manifolds
by
Yuka Taylor
The George Washington University
Coauthors: Chris Woodward, Rutgers University

As an application of the asymptotic formula of 6j symbols for the quantum enveloping algebra of sl(2), we construct a formal topological invariant of closed, oriented three-manifolds using spherical tetrahedra. We then explain how our spherical invariant predicts the semiclassical behavior of the Turaev-Viro invariant of three-manifolds. This is joint work with C. Woodward.

Date received: November 18, 2003


Non-left-orderable 3-manifold groups
by
Amir A. Togha
George Washington University
Coauthors: Mieczyslaw K. Dabkowski (UTD), Jozef H. Przytycki (GWU)

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched covers of S3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of S3 branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the 52 knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

Date received: December 3, 2003


Generalized Cochran sequence and a factorization of the Conway polynomial
by
Tatsuya Tsukamoto
Waseda University
Coauthors: Akira Yasuhara (Tokyo Gakugei University)

For a 2-component algebraically split link L, T.D. Cochran introduced a nortion ``derivative" of L and defined a sequence of link invariants. Then he showed that his sequence is equivarent to eta-function of L defined by S. Kojima and M. Yamasaki. We generalize these notions and results to a 3-component link L = K0 U J1 U J_2 with lk(Ji, K0)=0 (i=1, 2).

Date received: December 18, 2003


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