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Topology of Stein Manifolds
by
Selman Akbulut
MSU / IAS
Compact Stein manifolds are nothing but Lefschetz fibrations over a disk with bounded fibers and positive monodromy (PALF's). The constructive proof this will be given.
The surjective map PALF's -> Compact Stein manifolds is not 1-1 (for example B^4 has a unique Stein structure while it has many PALF structures corresponding to fibered knots in S^3). So in a sense PALF's are `primitives' of Stein structures (as chain complexes are `primitives' of homology groups). We will use the underlying PALF structures to give canonical compactifications of Stein manifolds to closed symplectic manifolds (no boundary). Examples will be discussed.
Date received: May 15, 2002
Approximating the coefficients of the HOMFLY polynomial by Vassiliev invariants
by
Laure Helme-Guizon
GWU
Not all knot invariants are Vassiliev invariants. An open question is whether any numerical knot invariant can be approximated by Vassiliev invariants, i.e. whether any numerical knot invariant is a pointwise limit of Vassiliev invariants.
It was proved that this conjecture holds in some special cases. For instance, Y Rong and I. Kofman found approximations by Vassiliev invariants for the coefficients of Jones polynomial.
In this talk, I will generalize this result by finding approximations by Vassiliev invariants for the coefficients of the Homfly polynomial.
Date received: May 16, 2002
Hyperbolic structure on a complement of tori in the 4-sphere
by
Dubravko Ivansic
The George Washington University
It is a familiar fact that many links in the 3-sphere have a complement that allows a hyperbolic structure. We generalize this phenomenon to one dimension higher by displaying a finite-volume noncompact hyperbolic 4-manifold M that is topologically the complement of 5 tori in the 4-sphere. The example stems from the work of J. Ratcliffe and S. Tschantz, who have used a computer to construct, via side-pairings of a polyhedron, many hyperbolic 4-manifolds. The manifold M is the orientable double cover of one of their manifolds. Using the construction of M and the fact that it is a complement in the 4-sphere we obtain a complicated Kirby diagram of the 4-sphere with dozens of 1-, 2- and 3-handles. While this diagram is equivalent via Kirby moves to the diagram of the standard differentiable 4-sphere, its complicatedness seems to suggest that any counterexample to the differentiable Poincare conjecture in dimension 4 given by a Kirby diagram could be very involved indeed.
Date received: May 20, 2002
Recent Results on Book Thickness of Graphs
by
Paul C. Kainen
Department of Mathematics, Georgetown University
The book thickness of a graph G is the chromatic number of the intersection graph formed by the (open) edges of G, minimized over all straightline drawings of G where the vertices lie on the unit circle. Shannon Overbay, in her 1998 thesis at Colorado State University, Fort Collins, showed that every bipartite planar graph has book thickness at most two through an argument based on Whitney's theorem (every planar triangulation with no separating triangles has a Hamiltonian cycle). We sketch a proof of Overbay's result and its applications.
Date received: May 16, 2002
Burnside groups in Knot Theory: 1. Solution to the Montesinos-Nakanishi 3-move conjecture
by
Jozef H. Przytycki
George Washington University
Coauthors: Mieczyslaw Dabkowski (GWU)
Yasutaka Nakanishi asked in 1981 whether a 3-move is an unknotting operation. In Kirby's problem list, this question is called ``The Montesinos-Nakanishi 3-move conjecture". We define the nth Burnside group of a link and use the 3rd Burnside group to answer Nakanishi's question; i.e. to show that some links cannot be reduced to trivial links by 3-moves. The conjecture has been proved to be valid for several classes of links: by Y.Nakanishi for links up to 10 crossings and Montesinos links, by J.Przytycki for links up to 11 crossings, Conway's algebraic links and closed 3-braids, by Q.Chen for links up to 12 crossings, closed 4-braids and closed 5-braids with the exception of the class of Gamma - the square of the center of the fifth braid group, by Przytycki and T.Tsukamoto for 3-algebraic links (including 3-bridge links), and Tsukamoto for (4, 5)-algebraic links (including 4-bridge links). Nakanishi presented in 1994, an example which he couldn't reduce by 3-moves: the 2-parallel of the Borromean rings (L2BR). We show that the links Gamma and L2BR are not 3-move reducible to trivial links.
Date received: May 15, 2002
Links and 3-manifolds with the same quantum invariants
by
Yongwu Rong
George Washington University and NSF
We present various constructions for links and 3-manifolds that share the same quantum invariants. These include classical mutation, our generalized mutations, more global mutations, and more recent constructions of non-trivial links with trivial Jones polynomial. Some ideas for further investigation will be discussed.
Date received: July 6, 2002
Copyright © 2002 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.