2006/02/13 by Józef H. Przytycki, Jozef H. Przytycki, Przytycki, Jozef H.
Mathematics · #57M25 (primary) #57M27 (secondary) #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA) #Rings, Modules, and Algebras #math.GT #math.QA #msc:57M25 #msc:57M27
paper · pdf · doi:10.48550/arxiv.math/0602264
58 pages, 44 figures, Chapter IX of the book "KNOTS: From combinatorics of knot diagrams to combinatorial topology based on knots
arxiv created 2006/02/13 · openalex publication_date 2006/02/13 · arxiv updated 2009/12/01 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
We describe in this chapter (Chapter IX) the idea of building an algebraic topology based on knots (or more generally on the position of embedded objects). That is, our basic building blocks are considered up to ambient isotopy (not homotopy or homology). For example, one should start from knots in 3-manifolds, surfaces in 4-manifolds, etc. However our theory is, until now, developed only in the case of links in 3-manifolds, with only a glance towards 4-manifolds. The main object of the theory is a skein module and we devote this chapter mostly to the description of skein modules in 3-dimensional manifolds. H. Poincare, in his paper "Analysis situs" (1895), abstractly defined homology groups starting from formal linear combinations of simplices, choosing cycles and dividing them by relations coming from boundaries The idea behind skein modules is to use links instead of cycles (in the case of a 3-manifold). More precisely we consider the free module generated by links modulo properly chosen (local) skein relations.