Skein modules of 3-manifolds

TL;DR

This paper introduces skein modules as a new algebraic invariant of 3-manifolds, aiming to interpret polynomial link invariants through algebraic topology and measure link isotopy obstructions.

Contribution

It proposes a novel algebraic invariant of 3-manifolds based on skein theory, extending the understanding of link isotopy and polynomial invariants.

Findings

Defines skein modules as algebraic invariants

Connects skein modules to link isotopy obstructions

Provides foundational framework for future research

Abstract

It is natural to try to place the new polynomial invariants of links in algebraic topology (e.g. to try to interpret them using homology or homotopy groups). However, one can think that these new polynomial invariants are byproducts of a new more delicate algebraic invariant of 3-manifolds which measures the obstruction to isotopy of links (which are homotopic). We propose such an algebraic invariant based on skein theory introduced by Conway (1969) and developed by Giller (1982) as well as Lickorish and Millett (1987). (This is the first paper I wrote about skein modules, almost 20 years ago. The recent survey of skein modules is available at http://arxiv.org/abs/math.GT/0602264.)