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Organizers |
Frobenius algebras, 2-dimensional surfaces ... and knots?
by
Lowell Abrams
George Washington University
A two-dimensional topological quantum field theory (tqft) provides a topological invariant of surfaces which is compatible with cutting and gluing operations. We describe a new characterization of Frobenius algebras, and use it to show that 2-d tqft's correspond to Frobenius algebras. We close with remarks directed towards the simple question "are Frobenius algebras useful in knot theory?"
Date received: May 9, 2001
Survey of Tits buildings
by
Jan Dymara
Ohio State University and Wroclaw University
Coauthors: Tadeusz Januszkiewicz
Definitions and examples of spherical, affine and Kac-Moody Tits buildings will be discussed. Buildings can be used to understand groups acting on them (e.g. affine buildings are acted upon by p-adic Lie groups). Later in this talk we will focus on representation theoretic properties of the group related to local spectral properties of the building.
Date received: May 8, 2001
Legendrian knots in overtwisted contact structures
by
Kasia Dymara
Ohio State University and Wroclaw University
We study the problem of classification of Legendrian knots in overtwisted contact structures on S3. The question is whether the two well-known integer-valued invariants (Thurston-Bennequin invariant and Maslov index), together with the knot type, form the full set of invariants. We give the positive answer in the case when there is an overtwisted disk intersecting none of the knots.
Date received: May 10, 2001
Rees Algebras
by
Sara Faridi
George Washington University
This talk is a survey of Rees Algebras and their applications within Commutative Algbera. I will give examples and discuss tools that one needs to compute theses rings.
Date received: May 9, 2001
The Lie algebra of curves on surfaces
by
Bill Goldman
University of Maryland
This talk will expound the Lie bialgebra based on oriented curves on an oriented surface. This Lie algebra was originally defined because of its representation in the Poisson algebra of the GL(n)-character variety of a compact surface. The Lie cobracket was discovered by Turaev. I will survey recent results on this Lie algebra and in particular discuss the relationship of acommutativity to disjointness.
Date received: May 8, 2001
A surgery description of an integral homology three-sphere respecting the Casson invariant
by
Makiko Ishiwata
Tokyo Woman's Christian University and George Washington University
In 1998, C.Lescop proved that any integral homology sphere with the Casson invariant zero can be obtained from S3 by surgery on a boundary link each component of which has trivial Alexander polynomial. In this talk, we show that for any integral homology sphere H, there exists an integer k such that H can be obtained from S3 by surgery on a boundary link each component of which has the Alexander polynomial 1+k(t1/2-t-1/2)2.
Date received: May 7, 2001
Skein modules, quantum deformations and string topology
by
Uwe Kaiser
Siegen University
| Abstract |
We discuss a general approach to understand skein modules of links in an oriented 3-manifold M from the following idea: Skein modules are part of deformed homology theories of suitable multi-loop spaces of M. In general, the multi-loop space is the space of differentiable maps of circles into M. The 0-th homology group of this space is an algebra of free homotopy classes of loops in M (classical observables). This is deformed into the skein modules using the stratification of the multi-loop space by singularities (Vassiliev theory), thus giving the transition to quantum observables. Nontrivial relations for the skein module appear from the string topology of M, e. g. suitable intersection pairings on the multi-loop space.
We discuss several explicit results for skein modules following from this approach: 1.) The universal Jones-Conway invariant (a generalization of the homflypt skein module), 2.) The quantum deformation of the fundamental group, a two-term skein module based on framed oriented links, and 3.) link homotopy skein modules (and their relation with recent work of Chas and Sullivan generalizing the Goldmann-Wolpert Lie algebra).
Date received: May 1, 2001
Infinite number of nontrivial links with trivial Jones polynomial
by
Louis H Kauffman
University of Illinois at Chicago
We discuss examples of nontrivial links with the trivial Jones polynomial. There is an infinite number of such examples being satellites of the Hopf link. The first such an example was noticed by M.Thistlethwaite, at the end of 2000, when he analysed 15 crossing links of two components. The question of V.F.R.Jones whether there is a nontrivial knot with the trivial Jones polynomial is still open. There is however a nontrivial virtual knot with the trivial Jones polynomial.
Date received: May 7, 2001
Skein theory for the Alexander Polynomial of 3-Manifolds via Hopf algebras
by
Thomas Kerler
The Ohio State University
We use the equivalence of the Frohman Nicas TQFT obtained via the cohomology of U(1)-moduli spaces and the Hennings TQFT for a certain 8-dimensional Hopf algebra in order to find a combinatorial desription of the Alexander polynomial of a 3-manifold that is presented as surgery along a link.
Date received: May 9, 2001
Invariants of Quantum Entanglement
by
Samuel J. Lomonaco, Jr.
University of Maryland Baltimore County
We discuss how Lie group invariants can be used to quantify the physical phenomenon of quantum entanglement. If time permits, we will discuss possible relationships with knot theory.
Date received: May 11, 2001
Seifert surgeries and the (-3, 3, 5) pretzel knot.
by
Thomas Mattman
CSU, Chico
Coauthors: Katura Miyazaki, Kimihiko Motegi
The (-3,3,5) pretzel knot is the first known example of a non-invertible knot which admits a Seifert fibered Dehn filling. We discuss the significance of this discovery and how the Seifert surgery can be exploited to determine the SL(2,C)-character variety of this knot.
Date received: April 17, 2001
Impossibility of obtaining split links from split links via twistings
by
Makoto Ozawa
Waseda University
We show that if a split link is obtained from a split link L in S3 by 1/n-Dehn surgery along a trivial knot C, then the link L ∪ C is splittable. That is to say, it is impossible to obtain a split link from a split link via a non-trivial twisting. As its corollary, we completely determine when a trivial link is obtained from a trivial link via a twisting.
Date received: April 24, 2001
Lagrangian approximation of local moves on links
by
Jozef H. Przytycki
George Washington University
There are various local moves on links which, conjecturally, can reduce any link to a trivial link, or more generally, any tangle to one of a finite collection of tangles (plus possibly trivial components). The moves, I will discuss, preserve the space of Fox n-colorings of a tangle T, for some n. For example a (p, q) move preserves Colpq+1(T). We use the fact that Fox p-colorings of a tangle yield a Lagrangian in the space of colorings of the boundary of the tangle, to approximate the number of tangles to which links can be reduced by the given moves.
Date received: May 9, 2001
Interactions between quantum topology and number theory
by
Adam S. Sikora
CRM Univ. Montreal
We will analyze some surprising connections between quantum invariants of knots and 3-manifolds and the number theory. In particular we will show how to use Kontsevich integral to study properties of zeta functions and we will relate the hyperbolic volume conjecture to the theory of modular forms. Also, if time permits, we will briefly describe a connection between the Jones polynomial and the Fermat's Last theorem.
Date received: May 9, 2001
Homological characterizations of planar and projective planar graphs.
by
Daniel Slilaty
The George Washington University
Coauthors: Lowell Abrams, The George Washington University
In the past, most characterizations of graphs that imbed in a given surface S have been Kuratowski-like forbidden subgraph characterizations; that is, a graph imbeds in S iff the graph does not contain a subdivision of one of a finite list of graphs FS. We will discuss some algebraic characterizations (old and new) of graphs that imbed in the plane or projective plane. These algebraic characterizations are inspired by classical homology theory.
Date received: May 3, 2001
The cubic skein module, Fox 7-colorings and unknotting number.
by
Mike Veve
George Washington University
We analyse the cubic skein module S4, \infty(M) = RLfr/(Sub) where (Sub) is the submodule generated by a cubic relation b0L0 + b1L1+ b2L2 + b3L3 + b\inftyL\infty and the framing relation L(1) = aL. We assume a, b0, b3 are invertible in the ring of coefficients R. For a manifold M being a 3-sphere we conjecture that the skein module is generated by trivial links. We analyse coefficients for which the unknot is a free generator and the (2, 3)-move is changing the sign of the skein element. We notice that the a (2, 3)-move is preserving the Fox 7-colorings of a link. We sketch the idea how to apply Traczyk's method of studying unknotting number with the help of our cubic skein module (or polynomial in a reduced case).
Date received: May 18, 2001
Ck-moves on spatial theta-curves and Vassiliev invariants
by
Akira Yasuhara
Tokyo Gakugei University and George Washington University
The Ck-equivalence is an equivalence relation generated by Ck-moves defined by Habiro. Habiro showed that the set of Ck-equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order≤ k-1. We see that the set of Ck-equivalence classes of the spatial \theta-curves forms a group under the vertex connected sum and that if the group is abelian, then it can be classified by the additive Vassiliev invariant of order≤ k-1. However the group is not necessarily abelian. In fact, we show that it is nonabelian for k ≥ 12. As an easy consequence, we have the set of Ck-equivalence classes of m-string links, which forms a group under the composition, is nonabelian for k ≥ 12 and m ≥ 2.
Date received: April 27, 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.