Finite type knot invariants and the calculus of functors

Abstract

We associate a Taylor tower supplied by the calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along the line of total degree zero leads to chord diagrams with relations as in finite type knot theory. We show that the spectral sequence collapses along this line and that the Taylor tower represents a universal finite type knot invariant.

On the Other Side of the Bialgebra of Chord Diagrams

“…In their approach the complexes are the first terms of the spectral sequences computing the cohomology groups of the homotopy fiber of the inclusion of the space of knots Emb = K \ Σ to the space of immersions Imm, cf. [37,25]. D. Sinha proved that this homotopy fiber is hopotopy equivalent to a direct product Emb × Ω 2 S d−1 , see [25].…”

“…Briefly speaking in this approach one "approximates" the space of knots Emb = K\Σ by means of homotopy limits of diagrams of maps. This approach was initiated by T. Goodwillie and M. Weiss [14,15], and then developped by D. Sinha [24,25], and also by I. Volic [37]. In particular for the space of long knots this method provides a spectral sequence whose second term is isomorphic up to a shift of bigradings to the first term of the Vassiliev spectral sequence (this spectral sequence was constructed by D. Sinha [24]): Proposition 0.1 (i) The groups of the first term of Vassiliev's spectral sequence (computing the (co)homology of the space of long knots) are naturally isomorphic up to a shift of bigradings to the groups of the second term of Sinha's spectral sequence (computing the (co)homology of a space homotopy equivalent to the space of long knots).…”

A Survey of Bott–taubes Integration

“…This extension was also used to place finite type knot invariants in a more homotopytheoretic framework [35]. We feel this work would benefit from a firmer grounding which we attempt to provide here.…”

On the homology of the space of knots

“…These results lead to some natural questions about the structure of the homology of the higher-dimensional embedding spaces K n;1 (n 4) studied recently by Sinha [37], Volic [41], Lambrechts [26] as well as others , CattaneoCotta-Ramusino-Riccardo [10], Goodwillie-Weiss [14], Kohno [24] and Sakai [33]). Constructions related to these questions are also addressed here.…”