Przytycki This project concerns the 11-year-old program of building an algebraic topology based on knots (or more generally on the position of embedded objects). That is, the basic building blocks are considered up to ambient isotopy (not homotopy or homology). For example, one starts from knots in 3-manifolds, surfaces in 4-manifolds, etc. This is a far-reaching program. Until now, one has been limited to 3-manifolds, with only a glance towards 4-manifolds. The principal objects of the theory are skein modules. One of the simplest skein modules is a quantization of the first homology group of a manifold. In general, skein modules of 3-manifolds are quotients of free modules over ambient isotopy classes of links by properly chosen local (skein) relations. The situation is somewhat reminiscent of that of ``classical'' algebraic topology 100 years ago (before Poincare's fundamental paper ``Analysis Situs,'' in 1895. Even three years ago one was able to compute only a few isolated examples, with the hope that the theory would rise in the future to a beautiful and powerful theory (the only exception was the Turaev-Przytycki construction of the Hopf algebra on the third skein module of the product of a surface and the interval). The situation has started to change in the last three years, and most of the progress concerns the Kauffman bracket skein module and its relation to character varieties of the fundamental group of a 3-manifold and the hyperbolic structure on a manifold. Topology, as foreseen by Leibniz in 1679, is the art of analyzing geometrical figures taking into account their position only; it does not take magnitudes into consideration (Leibniz called this art ``geometry situs''). In more modern language, topology describes spaces up to stretching and bending (but cutting and pasting is not allowed). Because of their flexibility, topological spaces are hard to analyze. In his fundamental paper of 1895, Analysis Situs, Henri Poincare associated topological spaces with algebraic objects that are invariant under space deformation (he called his objects homology and homotopy groups). The field of algebraic topology arose from the work of Poincare. Knot theory is the oldest branch of topology, first considered by A. Vandermonde in 1771. It studies the position of a circle (say a piece of rope with ends glued together) in space. Knot theory also has its roots in physics. W. Thomson (Lord Kelvin) proposed, in 1867, a theory of vortex atoms: that atoms were knotted tubes of ether. It was an aim of his friend P.G. Tait to describe the physical and chemical properties of particles in terms of the properties of related knots. Although the vortex theory was soon rejected, knot theory quickly developed to become an independent branch of topology. By the 1970's, some thought knot theory was out ot steam. Thus it came as a surprise when in 1984 Vaughan Jones discovered new algebraic invariants of knots (i.e., Jones polynomials). Jones' work was a breakthrough, providing solutions to old conjectures. In the same spirit as Leibniz, one would call the branch of mathematics that has its roots in Jones' construction, and Drinfeld's work on quantum groups, ``algebra situs.'' It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry, and statistical mechanics. ***

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Gerard A. Venema
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George Washington University
United States
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