Topology Atlas | Conferences

Abstracts

TQFT arising from subfactor
by
Marta Asaeda
University of Maryland

This talk is a survey of the relation between subfactor theory--a pair of certain C*-algebras with inclusion-- and the theory of the invariants of 3-mfds. I will talk mainly about the development after the discovery of Jones polynomial.

Date received: December 5, 2001

Interesting Kirby diagrams of the 4-sphere?
by
Dubravko Ivansic
The George Washington University

There exist hyperbolic manifolds obtained by pairing sides of a hyperbolic polyhedron that are complements of 5 tori in the 4-sphere. Using the construction of the hyperbolic manifold one can get a complicated Kirby diagram for the 4-sphere, with dozens of 1- and 2-handles. While this diagram is equivalent via Kirby moves to the diagram of the standard differentiable 4-sphere, its complicatedness seems to suggest that a counterexample to the differentiable Poincare conjecture in dimension 4 could be very involved indeed.

Date received: December 12, 2001

Knots, Vassiliev Invariants and Functional Integration without Integration
by
Louis H. Kauffman
University of Illinois at Chicago

There is a deep inteconnection between invariants of knots and links and the use of functional integrals as a heuristic (they do not exist). We replace the non-existent functional integral by a class of functions of gauge fields: defining F equivalent to G if F-G = DH where F,G,H are functions of a gauge field A that are "rapidly vanishing at infinity" in the sense that they go zero rapidly when an appropriate norm of A goes to infinity. DH denotes a functional derivative of F with respect to one of the gauge coordinates.We then define INT(F) to be the equivalence class of F. The talk will discuss how link invariants and Vassiliev invariants are intertwined with these INTegrals.

Date received: December 12, 2001

Identifications of Braid Group Representations
by
Thomas Kerler
The Ohio State Univeristy
Coauthors: Craig Jackson (Univeristy of Chicago)

This is a report on results emerging from the masters thesis of Craig Jackson, available at http://www.math.ohio-state.edu/ kerler/papers/craigthesis.ps

In the first part we consider the Krammer-Lawrence representation over Z[q, q^-1, t, t^-1] that has recently been proved faithful by Bigelow. We also consider the N-fold tensor product of the U_xi(sl₂) Verma module with heighets weight lambda (a generic complex number) and consider the R-matrix induced B_N-representation restricted to the subspace of sl₂-weight (N*lambda-4). We show both representations are equivalent if we identify t with xi^lambda and q with xi^-2.

In the second part we look at the stochastic representation of the string link semigroup over Q[t, t^-1] as proposed by Jones and constructed and investigated by Lin et al., which restricts to the unreduced Burau representations on braids. We identify the matrices over Q[t, t^-1] as ratios of two very naturally, skein theoretically defined representations over Z[t, t^-1], thus yielding an efficient way of computation.

Date received: December 12, 2001

Quantum Computing and Knot Theory
by
Samuel J. Lomonaco, Jr.
University of Maryland Baltimore County, Baltimore, MD

This talk begins with an introduction to quantum computing and concludes with knot theory

Date received: December 13, 2001

Rotors and the homology of branched double covers of links and tangles
by
Jozef H. Przytycki
George Washington University
Coauthors: Jan Dymara (Ohio State University and Wroclaw University), Tadeusz Januszkiewicz (Wroclaw University)

There is a classical result that a mutation of a link, L, preserves the 2-fold branched cover of S³ with L as the branching set, M⁽²⁾_L. In particular, H₁(M⁽²⁾_L;Z) is preserved by a mutation. We address the analogous problem for a rotation of a link. More precisely, we ask for which n and p (p a prime number), n-rotation preserves H₁(M⁽²⁾_L);Z_p. We prove that the answer is positive if there is an s such that p^s = -1  mod(n) or if n=p. For example 4-rotation preserves H₁(M⁽²⁾_L);Z₂₁) but we have examples when a 4-rotation changes H₁(M⁽²⁾_L);Z₅). In our approach, we interpret H₁(M⁽²⁾_L);Z_p) using Fox p-coloring space, Col_p(L). We introduce a symplectic structure on the Z_p^2n-2 (space of boundary colorings), and analyze Z_n action on it. From the fact that tangles yield Lagrangians in Z_p^2n-2 we are able to gain information whether Lagrangians invariant under Z_p-action are also invariant under dihedral (D_n) action.

Date received: December 11, 2001

3-Manifolds, Tangles and Persistent Invariant
by
Daniel S. Silver
University of South Alabama
Coauthors: Jozef H. Przytycki (GWU), Susan G. Williams (University of South Alabama)

Given a 2n-tangle t embedded in a link l, it is natural to ask which invariants of t necessarily persist as invariants of l. In his 1997 Ph.D. dissertation D. Krebes considered the case of a 4-tangle. He proved that if d divides the determinants of both the numerator closure and the denominator closure of the tangle, then d also divides the determinant of l. Since then two other proofs of Krebes's theorem have been given, one by D. Ruberman using classical algebraic topology and the other by Krebes, Silver and Williams using Temperley-Lieb algebra. Ruberman exploited a well-known relationship between the determinant of a link and the first homology group of its 2-fold cyclic branched cover. From his perspective, Krebes's theorem is a result about invariants of compact, oriented 3-manifolds that persist as invariants of rational homology 3-spheres in which they embed. We extend Ruberman's techniques in order to prove a generalization of Krebes's theorem for 2n-tangles. We also discuss results for the category of virtual links and tangles.

Date received: December 10, 2001

Torsion Numbers of Augmented Groups
by
Susan G. Williams
University of South Alabama
Coauthors: Daniel S. Silver (University of South Alabama)

Torsion and Betti numbers for knots are special cases of more general invariants b_r and \beta_r, respectively, associated to a finitely generated group G and epimorphism \chi: G → Z. The sequence of Betti numbers is always periodic; under mild hypotheses about (G, \chi), the sequence b_r satisfies a linear homogeneous recurrence relation with constant coefficients. Generally, b_r exhibits exponential growth rate. However, again under mild hypotheses, the p-part of b_r has trivial growth, for any prime p.

Date received: December 10, 2001

Copyright © 2001 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.