Research interests: Hugo Junghenn

Research interests: Jozef Przytycki
The classical knot theory studies a position of the circle (knot) or of several circles (link) in (3-dimensional) space. The fundamental problem of the classical knot theory is the classification of links (including knots) up to the natural movement in space which is called an ambient isotopy. To distinguish knots or links we look for invariants of links, that is, properties of links which are unchanged under ambient isotopy.

V. Jones started, in 1984, a revolution in the knot theory discovering a new powerful polynomial invariant of links.

My goal is to build an algebraic topology based on knots; that is a consistent theory in which links play the role of cycles, and skein modules the role of homology groups. Witten-Reshetikhin-Turaev-Wenzl invariants of 3-manifolds should correspond to some characteristic elements of cohomology groups. This is a far-reaching program. Until now we have been limited to 3-manifolds, with only a feeble glance towards 4-manifolds, and our skein modules generalize the first homology group and/or the fundamental group of a manifold (often being a quantization of the group). The situation is somehow reminiscent of that of ``classical" algebraic topology 100 years ago (before Poincaré's fundamental paper ``Analysis Situs", 1895). At present we are able to compute a few isolated examples, but I hope it will rise in future to a beautiful and powerful theory.

Selected Publications

Skein modules of 3-manifolds, Bull. Ac. Pol.: Math.; 39(1-2), 1991, 91-100 (Paper in which skein modules were introduced).
A survey of skein modules of 3-manifolds (with J. Hoste); in Knots 90, Proceedings of the International Conference on Knot Theory and Related Topics, Osaka (Japan), August 15-19, 1990, Editor A. Kawauchi, Walter de Gruyter 1992, 363-379.
Skein module of links in a handlebody, Topology 90, Proc. of the Research Semester in Low Dimensional Topology at OSU, Editors: B. Apanasov, W. D. Neumann, A. W. Reid, L. Siebenmann, De Gruyter Verlag, 1992; 315-342.
Quantum group of links in a handlebody, Contemporary Math: Deformation Theory and Quantum Groups with Applications to Mathematical Physics, M. Gerstenhaber and J. D. Stasheff, Editors, Volume 134, 1992, 235-245.
Vassiliev-Gusarov skein modules of 3-manifolds and criteria for periodicity of knots, Low-Dimensional Topology, Knoxville, 1992 ed.: Klaus Johannson International Press Co., Cambridge, MA 02238, 1994, 143-162.
n index of a graph with applications to knot theory (with K. Murasugi); Memoirs of the American Math. Soc., Vol. 106, Number 508, November 1993, 101 pages.