Topology Atlas | Conferences

 KNOTS in WASHINGTON XV (2nd Japan-USA Workshop in Knot Theory) January 10-15, 2003 George Washington University and Johns Hopkins University Washington, DC and Baltimore, MD, USA

Organizers
Kazuaki Kobayashi, Jozef H. Przytycki, Yongwu Rong, Shin-ichi Suzuki, Kouki Taniyama, Tatsuya Tsukamoto, Akira Yasuhara

# Abstracts

by
Charilaos Aneziris

By generalizing the Dowker notation to a notation suitable for multicomponent links, it is possible to create an algoritrhm that can be used for the Tabulation of Links. In this speech I discuss the main ideas behind this project, and present some preliminary results.

Representations of Tangle Categories and the Bracket Polynomial
by
John Armstrong
Yale University

In this talk, representations of categories of framed unoriented tangles which factor through Temperley-Lieb categories are described. These functors have the property that their restrictions to (0,0)-tangles (knots and links) are Bracket Polynomial evaluations. In particular, the "unitary" representations are calculated explicitly.

Fusion Rules of orbifold systems.
by
Marta Asaeda
University of Maryland

I will talk about the fusion algebras arising from orbifold systems of Uq(sl(n)) for q root of unity, especially about the fixed point resolution.

Achirality of Knots
by
M. Azram
IIUM, Kuala Lumpur 53100, Malaysia

A theoretic and diagrammatic relationship between knots and planar graphs has been established. It has been shown that in reduced alternating achiral knots, number of black regions is same as the number of white regions and consequently, W(K)=0 iff B = W is a necessary condition for reduced alternating knot to be achiral. It has been proved that if a reduced alternating achiral knot has p number of black regions then it has 2(p-1) crossings. This relationship enabled us to establish that the number of crossings and regions and hence, the number of vertices, edges and faces in the corresponding LR-Graphs are invariant. It has also been established that the number of vertices, edges and regions in LR-Graphs corresponding to black and white regions of reduced alternating achiral knot are same. A new way of constructing a reduced alternating achiral knot (linked link) has been suggested. Establishment of new but pivotal moves such as R*-move , 2(\pi)-twist and (\pi)-twist enabled us to prove that the black regions can be changed into white regions via Reidemeister moves. Consequently, the equivalence of the companion graphs, necessary and sufficient conditions for a reduced alternating knot to be achiral has been established.

Genome rearrangements and algebraic geometry
by
Baltimore

We develop a link between genome rearrangements and algebraic geometry . The evolutionary distance between genomes can be measured as a number of elementary rearrangements such as reversals. Such evolutionary changes can be modeled using algebraic geometry (i.e the combinatorics of some moduli spaces). Similar connections have been pointed out before by Waterman and Penner ( Adv. in Math. 1993), and by E.M.Jordan (1996). Our approach puts a bridge between Pevzner-Hannenhalli Theory, and the work of Davis et al., Devadoss and Yoshida. A simple application in topology provides a proof of the non-orientability of some spaces. The long range challenge is to apply these new understanding to study computational complexity of the problem.

Geometric Invariants for Knots in the Solid Torus
by
Khaled Bataineh
New Mexico State University

We study some geometric invariants for knots in the solid torus. Some of these invariants generalize geometric invariants for knots in the three-sphere and some of them are special for the solid torus. We give lower bounds on these geometric invariants using type one invariants for knots in the solid torus which are well understood and easy to calculate.

Concordance and rational knots
by
Jae Choon Cha
Indiana University

We continue the study of the structure of the concordance group of codimension two knots in rational homology spheres which was originated from Cochran and Orr, to give a full calculation. Using Seifert surfaces, we show that the rational knot concordance group is trivial in even dimensions, and is isomorphic to an algebraic rational concordance group that is defined to be a limit of ordinary algebraic concordance groups, in higher odd dimensions. We discover a complete set of invariants of the algebraic rational concordance group, and by calculating these invariants, we show that it is isomorphic to the sum of infinitely many copies of Z, Z/2, and Z/4. An investigation of norm subgroups on infinite towers of number fields using machinery of algebraic number theory plays a crucial role in calculating 4-torsion. We also show that the kernel and cokernel of the natural map of the ordinary concordance group into the rational concordance group, which measure the difference of them, are large enough to contain the sum of infinitely many copies of Z/2 and Z, Z/2, and Z/4, respectively. Our results can also be interpreted as a calculation of certain homology surgery obstruction gamma-groups.

The next simplest hyperbolic knots.
by
Abhijit Champanerkar
Columbia University
Coauthors: Ilya Kofman, Eric Patterson

While crossing number is the standard notion of complexity for knots, it is hard to compute. The number of ideal tetrahedra required to construct the complement provides a natural alternative. Callahan, Dean and Weeks determined knot complements with 6 or fewer tetrahedra in SnapPea's census of cupsed hyperbolic 3-manifolds and explicitly described the corresponding knots. We extend their hyperbolic knot census and identify all knots whose complements have 7 tetrahedra. We obtain 129 knot complements out of 3552 orientable, cusped hyperbolic 3-manifolds with 7 tetrahedra. Many of these simple'' hyperbolic knots have high crossing number. We also compute their Jones polynomials. This is joint work with Ilya Kofman and Eric Patterson

Non-left-orderable 3-manifold groups II
by
Mieczyslaw K. Dabkowski
The George Washington University
Coauthors: Ataollah Togha, Jozef H. Przytycki

We show that several torsion free 3-manifold groups are not left-orderable. Our examples arise from considering cyclic branched covers of S3 with links as branched sets. 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-left-orderable groups are obtained by taking 3-fold branched covers with various hyperbolic 2-bridge knots as branched sets. The manifold obtained in such a way from the 52 knot is of special interest since it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

On Legendrian knots
by
Oliver Dasbach
Louisiana State University

Legendrian knots are knots where the standard contact structure on R^3 vanishes. We will elaborate on the combinatorics of Legendrian knots.

Integral bases for TQFT Modules
by
Patrick Gilmer
Louisiana State University
Coauthors: Gregor Masbaum (Université Paris 7), Paul van Wamelen (Louisiana State University)

We construct integral bases for the SO(3)-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers.

Bordism Invariants of the Mapping Class Group
by
Aaron Heap
Rice University

We consider a 3-manifold M with an embedded, two-sided surface S in M. Let G denote the fundamental group of M and G_k denote the lower central series of G. Let g be a continuous map from M to the Eilenberg-MacLane space K(G/G_k, 1). Consider modifying M by removing a regular neighborhood of S and reattaching it via some surface homeomorphism f from the mapping class group, thus obtaining a new 3-manifold M(f). We also get a continuous map g(f) from M(f) to K(G/G_k, 1) by altering g. Let J(n) denote the generalized Johnson subgoup of the mapping class group of S. We show that if f is in J(2k-1) then (M(f), g(f)) and (M, g) are bordant over K(G/G_k, 1).

Approximation of the coefficients of the HOMFLYPT polynomial by Vassiliev invariants
by
Laure Helme-Guizon

A well-known conjecture states that any knot invariant is the pointwise limit of a sequence of Vassiliev invariants. This talk will prove that it is true in the special of the coefficients of the HOMFLYPT polynomial.

Finite Group Actions on Partially Peripheral 3-Manifolds
by
Toru Ikeda
Kochi Medical School

Brin, Johannson and Scott defined a 3-manifold M to be totally peripheral if every loop in M is freely homotopic into the boundary. We generalize this notion and define a 3-manifold M to be partially peripheral if every loop in M is a band sum of finite number of loops freely homotopic into the boundary. We study orientation-preserving finite group actions on a partially peripheral 3-manifold, which agree on the boundary, up to equivalence relative to the boundary.

A new construction of the Kauffman polynomial
by
Nikolai V. Ivanov
Michigan State University

We will present a new construction of the Kauffman polynomial (in its Dubrovnik version) inspired by X.-S. Lin's approach to finite type link invariants. A simpler version of this construction sheds a new light on the Kalfagianni-Lin approach to the 2-variable Jones (LYMPHTOFU) polynomial.

Determining the number of components of a link from its planar graph.
by
Jeff Johannes
SUNY Geneseo
Coauthors: Tim Nawojski, Holly Walrath

These are partial results of current research with undergraduates. In this talk we present techniques for computing the number of components of a link from its planar graph. We will discuss these results in the context of prior work with the planar graph of a knot or link.

On Zarankiewicz's conjecture regarding the crossing number of K_p, q
by
Paul C. Kainen
Department of Mathematics, Georgetown University

The crossing number of a graph is the least number of edge-intersections with which the graph can be drawn in the plane (subject to a few mild conditions). For the complete bipartite graph, the problem has an interesting history which will be briefly sketched, including a false proof in Zarankiewicz's original paper of a particular value for this crossing number. A new drawing scheme will be described which may show that the conjectured value is too large.

Deformation of string topology and skein modules of oriented 3-manifolds
by
Uwe Kaiser
Boise State University

Recently Moira Chas and Dennis Sullivan discovered interesting algebraic structures on the S1-equivariant homology of the space of maps from S1 into a d-dimensional smooth oriented manifold M. There is a natural approach of a geometric deformation of these structures for d=3. It replaces free homotopy classes of loops by isotopy classes of links and deforms coefficients at the same time. The relation of deformations with the theory of link homotopy skein modules is discussed. For the q-homotopy skein module naturally defined twisted string topology pairings are necessary. For isotopy skein modules the self-intersection component can be related to coalgebra structures on string homology and its generalizations.

Turaev-Viro invariant and spinal surfaces in a 3-manifold.
by
Joanna Kania-Bartoszynska
Boise State University
Coauthors: Charles Frohman (University of Iowa)

Turaev-Viro invariant of a 3-manifold can be expressed as a sum over spinal surfaces. This formulation leads itself to extending the definition of the invariant when the value of a complex parameter is not a root of unity. This is joint work with Charles Frohman.

New obstructions for doubly slicing knots
by
Taehee Kim
Indiana University, Bloomington

A knot is doubly slice if it is the intersection of a three sphere with a trivially embedded two sphere in a four sphere. The resulting knot splits the two sphere into two distinct slicing disks for the knot. Thus, the term "doubly slice". The definition goes back to Fox and Sumners in the 60's, who developed an initial obstruction theory. Although extensive efforts have been made to better understand this natural four manifold relation on knots, only elementary obstructions have been discovered.

In recent years, Cochran, Orr, and Teichner have gained a deeper understanding of classical topological knot concordance using von Neumann signatures and new Blanchfield duality pairings on knots. Similarly, we seek insights into doubly slice knots.

We develop a bi-sequence of new obstructions for a knot being doubly slice containing the classical obstructions as initial cases. Examples are constructed to illustrate the non-triviality of these obstructions at all levels. Here analytic signatures play a key role. Also we induce a bi-filtration of the double concordance group.

4-manifolds, slice links, and Dwyer's filtration
by
Slava Krushkal
University of Virginia

I will discuss the approach to slice links provided by 4-dimensional surgery. A related result concerns the Dwyer's filtration on second homology of 4-manifolds.

Some computations of Ohtsuki series for 3-manifolds
by
Ruth Lawrence
Hebrew University, Jerusalem
Coauthors: Nori Jacoby

We will present some computational evidence on the asymptotic structure of Ohtsuki series for integer homology spheres obtained from 3-manifolds presented as surgery around closures of certain simple rational tangles.

Homology of finite cyclic coverings of links and Massey products
by
Daniel Matei
University of Tokyo, Japan

Let L be a link in S\sp 3 and let p be a prime number. A cohomology class \xi in H\sp 1(S\sp 3 \L;Z/pZ) defines a covering M\sb\xi of the link complement. We relate the rank of H\sb 1(M\sb\xi;Z/pZ) with the numbers \nu\sb k, k ≥ 1 of linearly independent non-vanishing (k+1)-fold Massey products of the form < \xi, ..., \xi, \eta > , as \eta ranges over H\sp 1(S\sp 3 \L;Z/pZ).

Exceptional surgery and boundary slopes
by
Thomas Mattman
California State University, Chico
Coauthors: Masaharu Ishikawa (Tokyo Metropolitan University, Japan), Koya Shimokawa (Saitama University, Japan)

Let X be a norm curve in the SL(2, C)-character variety of a knot exterior M. Let t = || b || / || a || be the ratio of the Culler-Shalen norms of two distinct non-zero classes a, b in H_1( M, Z). We demonstrate that either X has exactly two associated strict boundary slopes, t and -t, or else there are strict boundary slopes r_1 and r_2 with |r_1| > t and |r_2| < t. As a consequence, we show that there are strict boundary slopes near cyclic, finite, and Seifert slopes. We also prove that the diameter of the set of strict boundary slopes can be bounded below using the Culler-Shalen norm of those slopes.

Constructing Algebraic Links and Tangles for Low Edge Numbers
by
Cynthia McCabe
University of Wisconsin - Stevens Point

A method is given for economically constructing any piecewise-linear algebraic link or tangle. This construction, which involves tree diagrams, gives a new upper bound for the edge number of the link. Furthermore, it realizes a minimal-crossing projection of the link. When combined with Conway's basic polyhedra, the tangle constructions lead to upper bounds for larger classes of links.

Detecting torsion in skein modules using Hochschild homology
by
Michael McLendon
Washington College

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the skein module of this 3-manifold.

Action of braid groups related to double branched covers
by
Jozef H. Przytycki
GWU
Coauthors: Mieczyslaw Dabkowski (GWU)

There is a classical result that the Burau representation of the 3-braid group reduces at t = −1 to the representation φ1: B3 → SL(2, Z) with the kernel generated by (σ1σ2)6 (that is the square of the center of B3). The Burau representation at t = −1 is know to be related to the action of a braid group on the homology of the double branch cover of a punctured disk. We consider here the generalization of the above construction to the action of a braid group on the graded Lie ring associated to the lower central series of the 2-generator free group. The lower central series of a group G (G1 = G, G2 = [G, G], ..., Gn = [Gn−1, G]) yields the associated graded Lie ring of the group: L = L1 ⊕ L2 ⊕ ... ⊕ Li ⊕ ... where Li = Gi/Gi+1. The Lie bracket in L corresponds to the group bracket [g, h] = g−1h−1gh. We computed the representation φi: B3 → aut(Li) for i ≤ 5 and in every case ker(φi) contains (σ1σ2)6. For example φ5 → SL(6, Z) is given by: φ51) =
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We speculate on the usefulness of these representations and compare to actions of B3 on Burnside groups, in particular to the fact that (σ1σ2)6 acts non-trivially on G/G4 for G being the Burnside group on two generators and exponent five.

Knots and Graphs
by
Freydoon Rahbarnia

In this papar, we investigate ways to use graphs to study knots. We begin by signed graphs of knot diagrams, and how it is related to the signed graph of its mirror image. And we study the knot associated a sign graph compare to the knot associated to the dual of the graph. Then we proceed to analyze the knots associated to cyclic graphs and knots associated to trees.

On the Sato-Levine invariant
by
Dušan Repovš
University of Ljubljana, Ljubljana, Slovenia

I shall present the main results of recent joint work with Peter M. Akhmetiev (Steklov Mathematical Institute, Moscow, Russia) and Joze Malesic (Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia) on the Sato-Levine invariant. I shall also discuss some related topics and present a few selected interesting open problems.

Construction of Braid Representations Through Hyperplane Arrangements
by
Ofer Ron
Hebrew University of Jerusalem, Giv'at Ram
Coauthors: Advised by Prof. Ruth Lawrence

We construct a fiber bundle F → E → Xn (where Xn is the configuration space of n ordered points) using the configuration space of a set of hyperplanes as the fiber, and provide a section. Using this we obtain a homomorphism Pn=\pi1(Xn) → \Aut(\pi1(F)) which we then translate into a homomorphism on the first homology (with a twisted local coefficient system) of the fiber, thus obtaining a representation of Pn.

We will present the complete construction of representations for Pn using the configuration space of n hyperplanes in complex (n-2) dimensional space, which may be conjugate to the Burau representation, and point out the difficulties in the computation of the possibly more interesting representation using the configuration space of n lines in complex 2 dimensional space.

Finite-type invariants based on doubled-delta moves
by
Ted Stanford
New Mexico State University
Coauthors: James Conant, Jacob Mostovoy

Given a move M (or set of moves) on knot or link diagrams, one can define finite-type invariants based on M. If M is a crossing change, then this is the usual notion of finite-type invariant. I will discuss invariants based on the doubled-delta move, which generates S-equivalence of knots (the equivalence generated by Seifert matrices). Some of the same results that one obtains in the usual case carry over to the doubled-delta case. For example, there is a theorem that says that the Ohyama-style definition of n-equivalence, which has been developed extensively by Gusarov and Habiro, gives the same equivalence classes of knots as the Vassiliev-style filtration on formal linear combinations of knots. Also, invariants based on doubled-delta moves are closely related to the loop filtration on chord diagrams introduced by Garoufalidis and Rozansky.

On the sum of external angles of a convex polyhedron
by
Kouki Taniyama
Waseda University
Coauthors: Kazuhiro Ichihara (Nara Women's University)

Let P be a convex polyhedron in an n-dimensional Euclidean space En. For a codimension one subspace V in En, let PV denote the image of P under the orthogonal projection to V. Note that this PV is a convex polyhedron in V. Let lV be the number of (n-2)-cells in PV. Let l be the average of lV where V varies over all codimension one subspaces in En. Then we show that the sum of all external angles of P is equal to l\pi.

Non-left-orderable 3-manifold groups I
by
Ataollah Togha
The George Washington University
Coauthors: Mieczyslaw K. Dabkowski, Jozef H. Przytycki

We show that several torsion free 3-manifold groups are not left-orderable. Our examples arise from considering cyclic branched covers of S3 with links as branched sets. 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-left-orderable groups are obtained by taking 3-fold branched covers with various hyperbolic 2-bridge knots as branched sets. The manifold obtained in such a way from the 52 knot is of special interest since it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

On the classification of closed Pseudo-Anosov 3-braids in the standard contact 3-space
by
Tatsuya Tsukamoto
Waseda University
Coauthors: W. Menasco

This talk is about the work in progress. Birman and Menasco has shown that there are non-transversally simple knot, that is, transversal knots which cannot be classified by their topological knot types and their Bennequin number by analizing immersed surface. In fact their topological knot types are closed Pseudo-Anosov closed 3-braids. In our work, we analize a branched surface related to such knots. In this talk, we introduce a new transversal isotopy invariant based on such branched surfaces.

AP Theory: A Discrete, Purely Group Theoretic TOE
by
H. E. Winkelnkemper
University of Maryland

Due to its ultimate discrete holography, its own analogue of Donaldson's theorem, its own non-skein knot and linking theory, as well as two different interacting, topology changing, non-local dynamics, the sheer metamathematical existence of AP Theory should already be considered, in the 4D compact case, as an abstract unification of String/M Theory and Loop Quantum Gravity Theory, as well as a rigorous intrinsic background independent, universal solution to the Maldacena conjecture.

Linking numbers in rational homology 3-spheres, cyclic branched covers and infinite cyclic covers
by
Akira Yasuhara
Tokyo Gakugei University
Coauthors: Jozef H. Przytycki (The George Washington University)

We study the linking numbers in a rational homology 3-sphere and in the infinite cyclic cover of the complement of a knot. They take values in Q and in Q(Z[t, t-1]) respectively, where Q(Z[t, t-1]) denotes the quotient field of Z[t, t-1]. It is known that the modulo-Z linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-Z[t, t-1] linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate 'modulo Z' and 'modulo Z[t, t-1]'. When the finite cyclic cover of the 3-sphere branched over a knot is a rational homology 3-sphere, the linking number of a pair in the preimage of a link in the 3-sphere is determined by the Goeritz/Seifert matrix of the knot.