
Organizers 
Computing Triangulations of Mapping Tori of Surface Homeomorphisms
by
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 humanreadable 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.
http://www.math.utah.edu/~brinkman/
Date received: February 24, 2000
Hyperbolic Structure on "link" complements in S^{4}
by
Dubravko Ivanšić
The George Washington University
It is well known that many link complements in S^{3} support a hyperbolic structure. Can the same be said for one dimension higher, i.e. inside S^{4}?
Previous work of this author has determined that if a finitevolume noncompact hyperbolic 4manifold M is to be considered a complement of a codimension2 submanifold A inside a closed 4manifold N, that is, if M=NA, then A is necessarily a union of flat 2manifolds, thus, tori and Klein Bottles.
The most interesting situation seems to be when N=S^{4}. We show that two nonorientable examples from a long list of hyperbolic 4manifolds constructed by Ratcliffe and Tschantz have double covers that are homeomorphic to complements of several tori inside S^{4}.
http://gwis2.circ.gwu.edu/~divansic
Date received: February 8, 2000
Coefficients of Homfly polynomial and Kauffman polynomial are not finite type invariants
by
Gyo Taek Jin
Korea Advanced Institute of Science and Technology
Coauthors: Jung Hoon Lee
We show that the integervalued 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
by
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 FennRourke, of a "quasilink" associate with the abstract link diagram. Moreover the notion of an abstract link diagram is easily generalized to 4dimensional case.
Seiichi Kamada's Home Page (at USA)
Date received: January 12, 2000
Boundary links which are not homotopically split
by
Kazuaki Kobayashi
Tokyo Woman's Christian University
We consider links in the 3sphere 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 (hsplit links).
2. boundary links (\partiallinks).
3. linkhomotopically trivial in the strong sense (strong htrivial).
Spetially hsplit links is a new class from the splitness property side. An hsplit link is a
\partiallink and also a strong htrivial link by definition. So if a given link is a \partial
link but not a strong htrivial, then it is not hsplit. Similarly if a given link is a strong
htrivial link but not a \partiallink, then it is not hsplit. In the first part of this talk
we shall give such examples. Next we will give an example of a non hsplit link which is strong
htrivial and \pariallink. There is no numerical invariant for such links distingushing from h
split links. So we will give some characteristic properties for hsplit 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. nullcobordant links.
By definitions if L is a split ribbon link, L is a ribbon link and if L is a ribbon link, it is
nullcobordant. And if L is a split ribbon link, it is hsplit. There are examples of hsplit
links which are not nullcobordant by calculating the signature of links.
Date received: February 7, 2000
Approximating Jones coefficients and other link invariants by Vassiliev invariants
by
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
by
Weiping Li
Oklahoma State University
In this talk, we show that the representation varieties of \pi_{1}(S^{2} \(S^{2} ∩ L)) (a link L in S^{3}) 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 sequivalence classes of stable parabolic bundles over S^{2} \(S^{2} ∩ 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).
http://www.math.okstate.edu/~wli/
Date received: January 24, 2000
Representations to finite groups and characteristic varieties
by
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 torsionfree 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 lowindex subgroups of G. We also interpret the distribution of the primeindex 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 P2reducible 3manifolds
by
Daniel Matignon
University of MarseilleProvence
Let X be the complement of a regular neighborhood of a knot in S^{3}, and let T be its torus boundary. If r is a slope on T, we denote by X(r) the 3manifold obtained by producing a rDehn surgery on T. We say that X(r) is reducible, or P^{2}reducible, if it contains an essential 2sphere, or a projective plane, respectively. In this case, we say that r is a reducible slope, or a P^{2}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 P^{2}reducible slopes is one.
Date received: January 13, 2000
Complements of hyperbolic 4braid knots contain no closed embedded totally geodesic surfaces
by
Hiroshi Matsuda
University of Texas at Austin / University of Tokyo
We will examine closed incompressible surfaces which are embedded in complements of 4braid knots.
Date received: January 21, 2000
Spaces of Polygonal Knots
by
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
C_{n}moves and polynomial invariants for links
by
Haruko A. Miyazawa
Tsuda College
In 1993 K. Habiro defined a new local move called a C_{n}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 C_{n}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 C_{n}move.
Date received: February 15, 2000
The Kauffman polynomial of order 1
by
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


Date received: January 18, 2000
Knots, links and Physics
by
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
by
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 f_{c}:c → S^{3} is given. A set {f_{c}  c ∈ C(G)} of embeddings is realizable if there is an embedding h:G → S^{3} such that the restriction map h_{c} is ambient isotopic to f_{c} for any c ∈ C(G). In this talk on six specified graphs G, we give a necessary and sufficient condition for a set {f_{c}  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 3move and t_3, bar t_4move conjectures
by
Jozef H. Przytycki
George Washington University
We present general ideas which can lead to a solution of MontesinosNakanishi 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 C_{n} = B_{n}/(\sigma_{i})^{3} is finite iff n≤ 5. Assion found two basic cases in which C_{n}/(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 \Delta^{5} = (\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4})^{5} be a generator of the center of the braid group, B_{5}. (1) ``Symplectic case". C_{n}/(\Delta^{10}) is a finite group. (2) ``Unitary case". C_{n}/(\Delta^{15}) is a finite group. We should remark that \Delta^{30}=1 in C_{5}, and C_{5}/\Delta^{5} is a simple group PSp(4, 3) (projective symplectic group).
Our approach to "moves" conjectures is via the following very concrete problem: (i) The 5tangle yielded by the 5braid \Delta^{10} is 3equivalent to the trivial 5braid tangle. (ii)] The 5tangle yielded by the 5braid \Delta^{15} is t_{3}, t̅_{4} equivalent to the trivial 5braid tangle.
Date received: January 23, 2000
Exchangeable braids
by
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 diskband generators of B_{n} (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 2polyhedra in 3manifolds
by
Dušan Repovš
University of Ljubljana (Ljubljana, Slovenia)
We shall present several results on special 2polyhedra and fake surfaces. In particular, RepovsSkopenkov theorem on resolving 2polyhedra by fake surfaces (resp. special 2polyhedra) and its application to a reduction of the Whitehead Asphericity Conjecture. We shall include the BrodskyRepovsSkopenkov work on thickenings of 2polyhedra. We shall also present MitchellPrzytyckiRepovs and CavicchioliLickorishRepovs results on spines of 3manifolds with boundary of genus 1 (resp. genus > 1).
Date received: December 17, 1999
On higher order link polynomials
by
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 LickorishRong, 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
by
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 2cocycles are assigned to crossings of classical knot diagrams that are colored by elements of a finite quandle, and 3cocycles are assigned to triple points of colored diagrams of knotted surfaces. To define the statesum 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 statesum invariants are used to prove noninvertibility of some twist spun torus knots. Some computational methods of quandle cohomology and the statesum invariants will be discussed. For example, quotient homomorphisms and colored knot diagrams are used to prove nontriviality of cohomology groups for some quandles.
Date received: January 18, 2000
Cyclic group actions on manifoldsinformal introduction
by
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/pZaction on H_1(F)? What is the relationship between the number of fixed circles in a Z/pZaction on a 3manifold M and the induced Z/pZaction 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
by
Maxim Sokolov
George Washington University
Coauthors: Jozef H. Przytycki (George Washington University)
We prove the following theorem:
Theorem.
If a closed orientable 3manifold M admits an action of a
cyclic group Z_{p} where p is an odd prime integer and the
fixed point set of the action is S^{1} then H_{1}(M; Z_{p}) ≠ Z_{p}. The result does not hold for p=2.
Date received: January 26, 2000
A survey of the 3manifold invariants derived from Hopf objects focusing on diagrammatic language.
by
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 3manifold 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 KauffmanRadford 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 (WittenReshetikhinTuraev, Lyubashenko, TuraevViro, ChungFukumaShapere, and BarrettWestbury) and some special cases. We will also cover: the completeness of the (diagrammatic) involutory Kuperberg invariant for prime, orientable, closed 3manifolds; 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 3manifolds due to Kuperberg and Sawin respectively.
Date received: February 4, 2000
Brunnian links are determined by their complements
by
Ted Stanford
United States Naval Academy
Coauthors: Brian Mangum (Barnard College, Columbia University)
If L_{1} and L_{2} are two Brunnian links with all pairwise linking numbers 0, then L_{1} and L_{2} are equivalent if and only if they have homeomorphic complements. In particular, this holds for all Brunnian links with at least three components. If L_{1} is a Brunnian link with all pairwise linking numbers 0, and the complement of L_{2} is homeomorphic to the complement of L_{1}, then L_{2} may be obtained from L_{1} 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.
http://front.math.ucdavis.edu/math.GT/9912006
Date received: January 14, 2000
On bridge numbers of composite ribbon knots
by
Toshifumi Tanaka
Graduate School of Mathematics, Kyushu University
Bleiler and EudaveMuñoz showed that composite ribbon number one knots have twobridge 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 twobridge knot, then K#K! is ribbon number one knot. Conversely, we show that if K_{0} and K_{1} are knots which are minimal with respect to ribbon concordance and K_{0}#K_{1} is a ribbon number one knot, then K_{0} is equivalent to K_{1}!.
Date received: January 6, 2000
Band description of knots and Vassiliev invariants
by
Kouki Taniyama
Tokyo Woman's Christian University
Coauthors: Akira Yasuhara (Tokyo Gakugei University and GWU)
In 1993 K. Habiro defined C_{k}move of oriented links and around 1994 he proved that two oriented knots are transformed into each other by C_{k}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 (k_{1}, ..., k_{m}). When k_{1}= ... = k_{m}=1 the invariant coincides with Vassiliev invariant of order < m in the usual sense. Let k=k_{1}+ ... +k_{m}. We show that two oriented knots are transformed into each other by C_{k}moves if and only if they have the same Vassiliev invariants of type (k_{1}, ..., k_{m}). 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 selfcontained.
Date received: January 17, 2000
SL(2, C) Representations of Tunnel Number One Knots, Examples
by
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 4algebraic links
by
Tatsuya Tsukamoto
the George Washington University
We study the forth skein module for the nalgebraic links. First, we show 3algebraic 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 4algebraic links and MontesinosNakanishi conjecture.
Date received: January 20, 2000
Open Problems in Billiard Knots
by
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 selfintersections inside some billiard room (a billiard room is 3manifold inside R^{3} 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 3mainfold. 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
by
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 a_{2}(K) ≠ a_{2}(M) there are infinitely many knots J_{m} such that the delta distance between J_{m} and M coincide with a_{2}(K)a_{2}(M) and whose Vassiliev invariants of order less than or equal to n coincide with that of K. Here a_{2}(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
by
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 k^{th} 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
Copyright © 2000 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.