The space of all knots

Abstract

Throughout this book we used the definition of finite type invariants based on the Vassiliev skein relation. This definition is justified by the richness of the theory based on it, but it may appear to be somewhat ad hoc . In fact, in Vassiliev's original approach the skein relation is a consequence of a rather sophisticated construction, which we are going to review briefly in this chapter. One basic idea behind Vassiliev's work is that knots, considered as smooth embeddings 葷 1 → 葷 3 , form a topological space K . An isotopy of a knot can be thought of as a continuous path in this space. Knot invariants are the locally constant functions on K ; therefore, the vector space of R -valued invariants, where R is a ring, is the cohomology group H 0 ( K , R ). We see that the problem of describing all knot invariants can be generalized to the following: Problem. Find the cohomology ring H * ( K , R ). There are several approaches to this problem. Vassiliev replaces the study of knots by the study of singular knots with the help of Alexander duality and then uses simplicial resolutions for the spaces of singular knots. This method produces a spectral sequence which can be explicitly described. It is not clear how much information about the cohomology of the space of knots is contained in it, but the zero-dimensional classes coming from this spectral sequence are precisely the Vassiliev invariants.