Fundamentals of Kauffman bracket skein modules

Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach algebra situs. When looking at the panorama of skein modules1, we see, past the rolling hills of homologies and homotopies, distant mountains the Kauffman bracket skein module, and farther off in the distance skein modules based on other quantum invariants. We concentrate here on the basic properties of the Kauffman bracket skein module; properties fundamental in further development of the theory. In particular we consider the relative Kauffman bracket skein module, and we analyze skein modules of Ibundles over surfaces. History of skein modules from my personal perspective I would like to use this opportunity, of informal presentation,2 to give my personal history of algebraic topology based on knots (a more formal account was given in [Pr-7]). In July 1986 I left Poland invited by Dale Rolfsen for a visiting position at UBC. In January of 1987, Jim Hoste gave a talk at the first Cascade Mountains Conference (in Vancouver) and described his work on multivariable generalization of the Jones-Conway ([HOMFLY][PT]) polynomial of links in S3. He was convinced that his construction works for 2 colors when the first color is represented only by a trivial component. He had already succeeded in the case of 2-component 2-bridge links. His method, following Nakanishi, was to analyze link diagrams in an annulus (the trivial component being z axis). We immediately noticed (with Jim) that the analogous construction for the Kauffman bracket polynomial has an easy solution [H-P-1]. In March This is an extended version of a part of the talk “Panorama of skein modules”, given at Low Dimensional Topology Conference; Madeira, Portugal, January, 1998. 2 I would like here to thank Hanna Nencka for a titanic task of organizing the Madeira’s conference.