Algebraic Topology Based on Knots (Series on Knots and Everything)

This invaluable book describes the idea of building an algebraic topology based on knots (or, more generally, on the position of embedded objects). The author's basic building blocks are thus considered up to ambient isotopy (not homotopy or homotopy). For example, one should start from knots in 3-manifolds, surfaces in 4-manifolds, etc.

H Poincare, in his paper "Analysis situs" (1895), defined abstractly homology groups starting from formal linear combinations of simplices, choosing cycles and dividing them by relations coming from boundaries. The present author repeats this construction in the case of 3-manifolds taking links instead of cycles. More precisely, he divides the free module generated by links by properly chosen (local) skein relations. He generalizes in this way the first homology group of the manifold. In the choice of relations he is guided by Jones type polynomial invariants of links in S(3). Thus even for S(3) he gets a nontrivial result. Several examples of skein modules are given, starting from the q-deformation of the homology group of a manifold. One of the examples relates the homotopy skein module of a surface times interval to the universal enveloping algebra of the Goldman -- Wolpert Lie algebra of curves on the surface. The author discusses a torsion in skein modules (for example, for connected sums). Finally, he speculates about a Van Kampen-Seifert type theorem for 3-manifolds (glued along surfaces) and the formulas calling TQFT.