Author: Jozef H. Przytycki

Preprint: GWUM-2001-1

Note: For Proceedings of The First International Workshop on Graphs -- Operads -- Logic, Cuautitlan, Mexico, March 12-16, 2001.  MSC-class: 57M25 (Primary) 57M27 (Secondary)

Abstract:
Talk 1: Open problems in knot theory that everyone can try to solve. Knot theory is more than two hundred years old; the first scientists who considered knots as mathematical objects were A.Vandermonde (1771) and C.F.Gauss (1794). However, despite the impressive grow of the theory, there are simply formulated but fundamental questions, to which we do not know answers. I will discuss in this talk several such open problems, describing in detail the 20 year old Montesinos-Nakanishi conjecture. Our problems lead to sophisticated mathematical structures (I will describe some of them in next two talks), but today's description will be absolutely elementary.

Talk 2: Lagrangian approximation of Fox $p$-colorings of tangles. We start from the description of Fox $p$-colorings and we discuss rational moves on tangles. The talk culminates in the introduction of the symplectic structure on the boundary of a tangle in such a way that tangles yields Lagrangians in the symplectic space.

Talk 3: Historical Introduction to Skein Modules. I discuss, in this talk, skein modules, or as I prefer to say more generally, algebraic topology based on knots. I would like to describe, in more detail, skein modules related to the (deformations) of 3-moves and the Montesinos-Nakanishi conjecture but first I will give the general definition and I will make a short tour of the world of skein modules.