Moduli spaces of knots

A good introduction to Vassiliev's singularity theory approach is math.GT/9707212, which gives an overview (not getting bogged down in technical details) of how to approach the cohomology of space knots by resolving its complement, the space of singular knots. More recent progress is surveyed well here, which starts to use analogies with complements of hyperplane arrangements. The computations arising from this approach were streamlined and carried further by Turchin in math.QA/0105140, as well as his later papers.

A reference for the de Rham theory approach to knot spaces is math.GT/9910139, where the authors define a map from complexes of Feynman diagrams to the de Rham complex of a knot space. Longoni has written notes explaining the connection to quantum field theory, available here.

The approach through three-manifold topology is investigated in the papers of Budney and Hatcher. In math.GT/0309427 Budney defines an action of the little two-cubes operad on spaces of long knots and shows that action is free in dimension three. He finishes determining the homotopy type of the space of long knots in dimension three in math.GT/0506524, using a detailed understanding of the JSJ decomposition of a knot complement.

A nice introductory paper to the connection between knot spaces and calculus of the embedding functor is D. Sinha, "Topology of spaces of knots", available at math.AT/0202287. In this paper, Sinha describes three equivalent models for knot spaces and constructs spectral sequences converging to the homotopy and cohomology of spaces of knots in codimension at least 3. The rational collapse of these spectral seqences is proved by P. Lambrechts, V. Turchin, and I. Volic, using Kontsevich's rational formality of the little cubes operad, summarized in a talk by Lambrechts in summer '05, available here.

It follows that the cohomology of spaces of knots in dimension n is the Hochschild homology of the nth Poisson operad. More about this can be found in V. Turchin "On the homology of the spaces of long knots", math.QA/0105140, and D. Sinha "Operads and knot spaces", math.AT/0407039. In the latter paper, Sinha also proves that each space of knots is a two-fold loop space using machinery of McClure and Smith.

For classical knots, I. Volic shows in "Finite type invariants and calculus of functors", math.AT/0401440, that an algebraic variant of the Taylor tower classifies rational finite type knot invariants. Also, R. Budney, J. Conant, K. Scannell, and D. Sinha begin to construct finite type invariants over the integers using embedding calculus in "New perspectives on self-linking", math.GT/0303034. The theory becomes remarkably geometric, leading to a count of lines intersecting a knot in four points.

Also available is the paper Calculus of the embedding functor and spaces of knots by Ismar Volic.