Mathematics – Geometric Topology
Scientific paper
2006-02-13
Mathematics
Geometric Topology
58 pages, 44 figures, Chapter IX of the book "KNOTS: From combinatorics of knot diagrams to combinatorial topology based on kn
Scientific paper
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.
No associations
LandOfFree
Skein modules does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Skein modules, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Skein modules will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-167260