Topology Atlas | Conferences

KNOTS in WASHINGTON 10 Japan-USA; workshop in Knot Theory
January 23-30, 2000
George Washington University
Washington, DC, USA

Kazuaki Kobayashi, Jozef H. Przytycki, Yongwu Rong, Kouki Taniyama, Tatsuya Tsukamoto, Akira Yasuhara

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Computing Triangulations of Mapping Tori of Surface Homeomorphisms
Peter Brinkmann
University of Utah

I will present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomorphisms, suitable for Jeff Weeks's program SnapPea. The package consists of two programs. One of them computes triangulations and prints them in a human-readable format. The other one converts this format into SnapPea's triangulation file format and may be of independent interest because it allows for quick and easy generation of input for SnapPea.

Date received: February 24, 2000

Hyperbolic Structure on "link" complements in S4
Dubravko Ivanšić
The George Washington University

It is well known that many link complements in S3 support a hyperbolic structure. Can the same be said for one dimension higher, i.e. inside S4?

Previous work of this author has determined that if a finite-volume noncompact hyperbolic 4-manifold M is to be considered a complement of a codimension-2 submanifold A inside a closed 4-manifold N, that is, if M=N-A, then A is necessarily a union of flat 2-manifolds, thus, tori and Klein Bottles.

The most interesting situation seems to be when N=S4. We show that two nonorientable examples from a long list of hyperbolic 4-manifolds constructed by Ratcliffe and Tschantz have double covers that are homeomorphic to complements of several tori inside S4.

Date received: February 8, 2000

Coefficients of Homfly polynomial and Kauffman polynomial are not finite type invariants
Gyo Taek Jin
Korea Advanced Institute of Science and Technology
Coauthors: Jung Hoon Lee

We show that the integer-valued knot invariants appearing as the coefficients of the HOMFLY polynomial and the Kauffman polynomial are not of finite type.

Date received: February 28, 2000

Abstract link diagrams and virtual knots
Seiichi Kamada
Osaka City University/ University of South Alabama
Coauthors: Naoko Kamada (University of South Alabama)

The notion of an abstract link diagram was announced by N. Kamada at a regional conference in 1993 and an international conference in 1996 held at Waseda University. We show that this notion is equivalent to Kauffman's virtual knots. Then each of upper and lower presented quandles of a virtual knot (or of its corresponding abstract link diagram) has a geometrical meaning, which can be interpreted as the fundamental quandle, in the sense of Joyce and Fenn-Rourke, of a "quasi-link" associate with the abstract link diagram. Moreover the notion of an abstract link diagram is easily generalized to 4-dimensional case.

Seiichi Kamada's Home Page (at USA)

Date received: January 12, 2000

Boundary links which are not homotopically split
Kazuaki Kobayashi
Tokyo Woman's Christian University

We consider links in the 3-sphere and consider them from the splitness property side. The geometrically split has the strongest splitness property but it is a "product" of knots. In this talk consider the following three kinds of links ,
        1. homotopically split links (h-split links).
        2. boundary links (\partial-links).
        3. link-homotopically trivial in the strong sense (strong h-trivial).
Spetially h-split links is a new class from the splitness property side. An h-split link is a \partial-link and also a strong h-trivial link by definition. So if a given link is a \partial- link but not a strong h-trivial, then it is not h-split. Similarly if a given link is a strong h-trivial link but not a \partial-link, then it is not h-split. In the first part of this talk we shall give such examples. Next we will give an example of a non h-split link which is strong h-trivial and \parial-link. There is no numerical invariant for such links distingushing from h- split links. So we will give some characteristic properties for h-split links and give an example. There is another sequence of links with respest to splitness property side as followings.
        4. split ribbbon links.
        5. ribbon links.
        6. null-cobordant links.
By definitions if L is a split ribbon link, L is a ribbon link and if L is a ribbon link, it is null-cobordant. And if L is a split ribbon link, it is h-split. There are examples of h-split links which are not null-cobordant by calculating the signature of links.

Date received: February 7, 2000

Approximating Jones coefficients and other link invariants by Vassiliev invariants
Ilya S. Kofman
University of Maryland, College Park
Coauthors: Yongwu Rong (George Washington University)

We find approximations by Vassiliev invariants for the coefficients of the Jones polynomial and all specializations of the HOMFLY and Kauffman polynomials. Consequently, we obtain approximations of some other link invariants arising from the homology of branched covers of links.

Date received: January 24, 2000

Link invariant from representation variety
Weiping Li
Oklahoma State University

In this talk, we show that the representation varieties of \pi1(S2 \(S2 ∩ L)) (a link L in S3) with different conjugacy classes in SU(2) along meridians are symplectic stratified varieties from the group cohomology point of view. The variety can be identified with the moduli space of s-equivalence classes of stable parabolic bundles over S2 \(S2 ∩ L) with corresponding weights along punctures, and also can be identified with the moduli space of gauge equivalence classes of SU(2)-flat connections with prescribed holonomies along punctures. We obtain an invariant of links (knots) from intersection theory on such a moduli space (a generalization of the signature of the link).

Date received: January 24, 2000

Representations to finite groups and characteristic varieties
Daniel Matei
University of Rochester
Coauthors: Alex Suciu (Northeastern University)

In a paper from 1935, Philip Hall introduced an invariant of finitely presented groups that counts representations onto finite groups. Let G be a finitely presented group with torsion-free abelianization (for example a link group). Following an idea of Fox, we compute Hall's invariant for certain metabelian representations in terms of the characteristic varieties of the group G. These varieties are defined by the Alexander ideals of G. As an application, we count the number of low-index subgroups of G. We also interpret the distribution of the prime-index normal subgroups of G, according to their abelianization, in terms of the characteristic varieties of G.

Date received: January 13, 2000

Dehn surgeries and reducible, or P2-reducible 3-manifolds
Daniel Matignon
University of Marseille-Provence

Let X be the complement of a regular neighborhood of a knot in S3, and let T be its torus boundary. If r is a slope on T, we denote by X(r) the 3-manifold obtained by producing a r-Dehn surgery on T. We say that X(r) is reducible, or P2-reducible, if it contains an essential 2-sphere, or a projective plane, respectively. In this case, we say that r is a reducible slope, or a P2-reducible slope. The distance between two distinct slopes is the geometric minimal number of intersection between them.

The results of this talk are that the distance between two reducible slopes, or between two P2-reducible slopes is one.

Date received: January 13, 2000

Complements of hyperbolic 4-braid knots contain no closed embedded totally geodesic surfaces
Hiroshi Matsuda
University of Texas at Austin / University of Tokyo

We will examine closed incompressible surfaces which are embedded in complements of 4-braid knots.

Date received: January 21, 2000

Spaces of Polygonal Knots
Kenneth C. Millett
University of California, Santa Barbara

The structure of the spaces of polygonal knots will be discussed from several perspectives: geometric, physical, statistical and computational. The basic structures will be described in relationship to current knowledge, new results, and interesting conjectures suggested them.

Date received: January 7, 2000

Cn-moves and polynomial invariants for links
Haruko A. Miyazawa
Tsuda College

In 1993 K. Habiro defined a new local move called a Cn-move. It is known that this local move is closely related to Vassiliev invariants, that is, if oriented links L and L' are transformed into each other by Cn-moves, then they have the same value of any Vassiliev invariant of order less than n. In this talk we study the difference of some Vassiliev invariants of order n for two links L and L' which are transformed into each other by a single Cn-move.

Date received: February 15, 2000

The Kauffman polynomial of order 1
Yasuyuki Miyazawa
Yamaguchi University

In 1997, Y. Rong defined a polynomial invariant of a link which is related to the Homfly polynomial and Vassiliev invariant (of order 1). The invariant is called the first order skein (or Homfly) polynomial of a link. It satisfies the skein relations
PL×=xPL++yPL-+zPL0     and     PL××=0.
In this talk, we study a regular isotopy invariant H satisfying the following skein relations
HL×=xHL++yHL-+zHL0+zHL\infty     and     HL××=0.
(In some sense, H may be thought of as the first order Kauffman polynomial of a link.) In order to do that, we will introduce a magnetic graph for a definition of the Kauffman polynomial (of order 0) of a link. Using the graph and the definition, we will define the Kauffman polynomial of order 1 for a link.

Date received: January 18, 2000

Knots, links and Physics
Michael Monastyrsky
Institute of Theoretical and Experimental Physics , Moscow 117259 ,Russia

In our talk we discuss the relation between Physics and Topology, especially knot theory. We start from the first topological work of Leibnitz "Geometrica Dedicatica" and go through with the recent applications of the knot theory to condensed matter including classifications of linked defects in liquid crystals and superfluid liquids.

Date received: February 25, 2000

A characterization of knots in a spatial graph
Kazuko Onda
Graduate School of Mathematics, Tsuda College

For a finite graph G, let C(G) be the set of all cycles of G. Suppose that for each c ∈ C(G), an embedding fc:c → S3 is given. A set {fc  |  c ∈ C(G)} of embeddings is realizable if there is an embedding h:G → S3 such that the restriction map h|c is ambient isotopic to fc for any c ∈ C(G). In this talk on six specified graphs G, we give a necessary and sufficient condition for a set {fc  |  c ∈ C(G)} to be realizable by using second coefficient of Conway polynomial of knot.

Date received: January 20, 2000

Symplectic and unitary quotients of Burau representation, and 3-move and t_3, bar t_4-move conjectures
Jozef H. Przytycki
George Washington University

We present general ideas which can lead to a solution of Montesinos-Nakanishi conjecture and its generalizations. In particular we speculate about the geometrical meaning of the finite quotients of the braid group described by J.Assion (Symplectic and Unitary cases). Coxeter showed that Cn = Bn/(\sigmai)3 is finite iff n≤ 5. Assion found two basic cases in which Cn/(Ideal) is finite: ``Symplectic and Unitary" cases. We noticed with B.Westbury (motivated by Q.Chen) simple presentations for Assion ideals (compare also Murasugi). Let \Delta5 = (\sigma1\sigma2\sigma3\sigma4)5 be a generator of the center of the braid group, B5. (1) ``Symplectic case". Cn/(\Delta10) is a finite group. (2) ``Unitary case". Cn/(\Delta15) is a finite group. We should remark that \Delta30=1 in C5, and C5/\Delta5 is a simple group PSp(4, 3) (projective symplectic group).

Our approach to "moves" conjectures is via the following very concrete problem: (i) The 5-tangle yielded by the 5-braid \Delta10 is 3-equivalent to the trivial 5-braid tangle. (ii)] The 5-tangle yielded by the 5-braid \Delta15 is t3, t̅4 equivalent to the trivial 5-braid tangle.

Date received: January 23, 2000

Exchangeable braids
Marta Rampichini
Dipartimento di Matematica dell'Università Statale di Milano, Italy
Coauthors: Hugh Morton (Liverpool, UK), Maria Dedò (Milano, Italy)

Two links A, B are exchangeably braided if each of them is a (generalized) braid relative to the other. This situation can be described by a finite set of combiantorial data, extracted from the singular foliation induced by the fibration of B on each fibre of A (or viceversa). If one of the two links is the unknot, then the other one is a classical braid. To express it by a word in the disk-band generators of Bn (cf Birman, Ko, Lee) allows to find an algorithm to identify exchangeable braids. Isotopies of fibres are so translated into conjugations and relations of braid words, with a nice connection between topology and algebra.

Date received: January 20, 2000

Topology of 2-polyhedra in 3-manifolds
Dušan Repovš
University of Ljubljana (Ljubljana, Slovenia)

We shall present several results on special 2-polyhedra and fake surfaces. In particular, Repovs-Skopenkov theorem on resolving 2-polyhedra by fake surfaces (resp. special 2-polyhedra) and its application to a reduction of the Whitehead Asphericity Conjecture. We shall include the Brodsky-Repovs-Skopenkov work on thickenings of 2-polyhedra. We shall also present Mitchell-Przytycki-Repovs and Cavicchioli-Lickorish-Repovs results on spines of 3-manifolds with boundary of genus 1 (resp. genus > 1).

Date received: December 17, 1999

On higher order link polynomials
Yongwu Rong
George Washington University

Higher order link polynomials were defined by combining ingredients from link polynomials and Vassiliev invariants. In this talk, we will survey the following results on this topic:

1. The classification of the order 1 Homfly polynomial, done by the speaker.

2. The theorem that each nth partial derivatives of the Homfly polynomials is a higher order Homfly polynomial of order n, due to Lickorish and the speaker. This also greatly simplifies the work in (1).

3. The determination of the free part of the higher order Conway skein module, due to Andersen and Turaev.

4. An affirmative answer to the question, asked by Lickorish-Rong, whether all partial derivatives of the Homfly link polynomials are linearly independent.

5. The classification of all the higher order Conway polynomials, following work of the above.

Date received: January 24, 2000

Some geometric aspects of quandle (co)homology and cocycle invariants of knotted curves and surfaces
Masahico Saito
University of South Florida
Coauthors: J.S. Carter, D. Jelsovsky, S. Kamada, L. Langford

A cohomology theory of quandles is defined and applications to knot theory will be given. Cocycle conditions are derived from Reidemeister moves, and generalized to a cochain complex. Quandle 2-cocycles are assigned to crossings of classical knot diagrams that are colored by elements of a finite quandle, and 3-cocycles are assigned to triple points of colored diagrams of knotted surfaces. To define the state-sum invariants, the product of all cocycles assigned to crossings is computed, and the sum is taken over all possible colorings of a given knot diagram. The state-sum invariants are used to prove non-invertibility of some twist spun torus knots. Some computational methods of quandle cohomology and the state-sum invariants will be discussed. For example, quotient homomorphisms and colored knot diagrams are used to prove non-triviality of cohomology groups for some quandles.

Date received: January 18, 2000

Cyclic group actions on manifolds-informal introduction
Adam S. Sikora
Univ. of Maryland

Let a cyclic group Z/pZ act on a surface F. Is the number of fixed points of this action determined by the induced Z/pZ-action on H_1(F)? What is the relationship between the number of fixed circles in a Z/pZ-action on a 3-manifold M and the induced Z/pZ-action on H_1(M)? We will answer these and analogous questions using equivariant (Tate) cohomology, and a careful analysis of differentials in associated spectral sequences.

Date received: February 26, 2000

Surgeries on periodic links and homology of periodic manifolds
Maxim Sokolov
George Washington University
Coauthors: Jozef H. Przytycki (George Washington University)

We prove the following theorem:

Theorem. If a closed orientable 3-manifold M admits an action of a cyclic group Zp where p is an odd prime integer and the fixed point set of the action is S1 then H1(M; Zp) ≠ Zp. The result does not hold for p=2.

Maxim Sokolov's math research

Date received: January 26, 2000

A survey of the 3-manifold invariants derived from Hopf objects focusing on diagrammatic language.
Fernando J. O. de Souza
Los Alamos National Laboratory/ University of IL at Chicago

The research in quantum topology led to the construction of several 3-manifold invariants by means of either some kinds of Hopf algebras, or their categories of representations, or Hopf objects (objects in categories with reasonable structures, endowed with morphisms that satisfy the Hopf axioms as in the case of Hopf algebras.) This survey explores these invariants, particularly the involutory Kuperberg invariant and the Kauffman-Radford reformulation of the Hennings (KRH) invariant.

On the one hand, they can be seen as particular cases of other invariants when they are defined via some Hopf algebras. On the other hand, they can be defined at the general level of Hopf objects by means of diagrams representing morphisms. We will start by reviewing the effect of several categorical structures on the typical representation of objects (resp. morphisms) as edges (resp. vertices) of planar immersions of graphs. We will also demonstrate the use of diagrams to represent morphisms of Hopf objects modulo Hopf axioms, and recall the realization of this diagrams in Hopf algebras. We will then build the involutory Kuperberg and KRH invariants, and review their relations with other invariants (Witten-Reshetikhin-Turaev, Lyubashenko, Turaev-Viro, Chung-Fukuma-Shapere, and Barrett-Westbury) and some special cases. We will also cover: the completeness of the (diagrammatic) involutory Kuperberg invariant for prime, orientable, closed 3-manifolds; and the KRH invariant on a diagrammatic, Drinfeld quantum double of any involutory Hopf object (conjectured to be equivalent to the involutory Kuperberg invariant.) Finally, we will mention their generalizations to framed 3-manifolds due to Kuperberg and Sawin respectively.

Date received: February 4, 2000

Brunnian links are determined by their complements
Ted Stanford
United States Naval Academy
Coauthors: Brian Mangum (Barnard College, Columbia University)

If L1 and L2 are two Brunnian links with all pairwise linking numbers 0, then L1 and L2 are equivalent if and only if they have homeomorphic complements. In particular, this holds for all Brunnian links with at least three components. If L1 is a Brunnian link with all pairwise linking numbers 0, and the complement of L2 is homeomorphic to the complement of L1, then L2 may be obtained from L1 by a sequence of twists around unknotted components. These results lead to a straightforward way of reducing the problem of detecting a trivial link to the problems of detecting and straightening out a trivial knot.

Date received: January 14, 2000

On bridge numbers of composite ribbon knots
Toshifumi Tanaka
Graduate School of Mathematics, Kyushu University

Bleiler and Eudave-Muñoz showed that composite ribbon number one knots have two-bridge summands. We show that there exists an infinite family of composite ribbon number one knots which have arbitrary large bridge numbers. If K is a two-bridge knot, then K#-K! is ribbon number one knot. Conversely, we show that if K0 and K1 are knots which are minimal with respect to ribbon concordance and K0#K1 is a ribbon number one knot, then K0 is equivalent to -K1!.

Date received: January 6, 2000

Band description of knots and Vassiliev invariants
Kouki Taniyama
Tokyo Woman's Christian University
Coauthors: Akira Yasuhara (Tokyo Gakugei University and GWU)

In 1993 K. Habiro defined Ck-move of oriented links and around 1994 he proved that two oriented knots are transformed into each other by Ck-moves if and only if they have the same Vassiliev invariants of order < k. However this deep theorem appears only in his recent paper that develops his original clasper theory. In this talk we define Vassiliev invariant of type (k1, ..., km). When k1= ... = km=1 the invariant coincides with Vassiliev invariant of order < m in the usual sense. Let k=k1+ ... +km. We show that two oriented knots are transformed into each other by Ck-moves if and only if they have the same Vassiliev invariants of type (k1, ..., km). As a corollary we have Habiro's Theorem. Our proof is based on a concept which we call band description of knots. Our proof is elementary and completely self-contained.

Date received: January 17, 2000

SL(2, C) Representations of Tunnel Number One Knots, Examples
Debora M. Tejada
Universidad Nacional de Colombia - University of Texas at San Antonio
Coauthors: Hugh M. Hilden (University of Hawaii at Honolulo), Margarita M. Toro (Universidad Nacional de Colombia)

We prove that the knot group of a tunnel number one knot has presentation with two generators and a palindrome as relation. We also compute the SL(2, C) representations of groups that have this kind of presentation. We give some examples.

Date received: February 26, 2000

The forth skein module for 4-algebraic links
Tatsuya Tsukamoto
the George Washington University

We study the forth skein module for the n-algebraic links. First, we show 3-algebraic links are generated as a linear combination of trivial links, which is a joint work with J. Przytycki. And second, we study the forth skein module for the 4-algebraic links and Montesinos-Nakanishi conjecture.

Date received: January 20, 2000

Open Problems in Billiard Knots
Michael A. Veve
George Washington University

While mathematical billiards have been studied quite extensively for many years the same cannot be said for billiard knots. Billiard knots are a special type mathematical billiard, namely, periodic trajectories without self-intersections inside some billiard room (a billiard room is 3-manifold inside R3 with a piecewise smooth boundary). As the terminology suggests, the study of billiard knots is primarily concerned with how the periodic orbit is knotted inside the 3-mainfold. We discuss some open problems concerning billiard knots and report on some of the progress made in solving these problems.

Date received: January 20, 2000

Delta distance and Vassiliev invariants of knots
Harumi Yamada
Tokyo Woman's Christian University

It is shown by Y. Ohyama, K. Taniyama and S. Yamada that for any natural number n and any knot K, there are infinitely many unknotting number one knots whose Vassiliev invariants of order less than or equal to n coincide with that of K. We analize it for delta unknotting number and obtain the following. For any natural number n and any oriented knots K and M with a2(K) ≠ a2(M) there are infinitely many knots Jm such that the delta distance between Jm and M coincide with |a2(K)-a2(M)| and whose Vassiliev invariants of order less than or equal to n coincide with that of K. Here a2(K) is the second coefficient of the Conway polynomial of K.

Paper reference: doi:10.1142/S0218216500000566

Date received: February 29, 2000

Vassiliev Theory as Deformation Theory
David N. Yetter
Dept. of Mathematics, Kansas State University

It would appear to be fruitful to consider Vassiliev invariants of knots, links, and tangles in the context of an algebraic deformation theory for braided monoidal categories.

We describe the construction of cochain complexes associated to monoidal categories, monoidal functors, and braided monoidal categories, and theorems relating the cohomology of the category (functor) to infinitesimal deformations of its structure maps.

When a symmetric monoidal category with duals is deformed to give rise to a ribbon category, the kth term in the functorial link invariant associated to the deformed category is seen to be a Vassiliev invariant of degree≤ k.

Ëxtrinsic deformations" in which a braided monoidal category is deformed a subcategory of a larger category are shown to provide a setting for the consideration of universal Vassiliev invariants over general coefficient rings.

Date received: February 28, 2000

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