Moduli spaces of knots

January 3 to January 7, 2006

at the

American Institute of Mathematics, Palo Alto, California

organized by

Fred Cohen, Allen Hatcher, and Dev Sinha

Original Announcement

This workshop will be devoted to the study of the global topology of spaces of embedded curves in Euclidean spaces and other manifolds. This topic has recently promoted a renewed vigorous interchange between algebraic and geometric topology. There are four basic, fundamentally different approaches to the study of these spaces of knots:
  1. Singularity theory: studying the space of knots through its complement, the space of singular knots. Pioneered by Vassiliev, this led to the study of finite-type knot invariants.
  2. de Rham theory and quantum field theory: A knot's "energy" is "integrated over all connections in perturbative expansion" to define invariants. This approach has subsequently been recast in classical de Rham theory, tying in to theory of Chen integrals, and extended to higher dimensions.
  3. Calculus of embeddings: A vast generalization of Hirsch-Smale theory, the calculus gives successively better approximations to spaces of embeddings. In the setting of knots, the information can be encoded using compactified configuration spaces.
  4. Three-manifold techniques: In dimension three it is feasible to study the space of embeddings of a knot by comparing the space of diffeomorphisms of the ambient manifold with the space of diffeomorphisms of the knot�s complement.
All of these approaches have been enriched by use of the little cubes operad, in various roles and guises.

This workshop will bring together practitioners of each of these approaches for the first time, along with researchers in related fields, to work together on the important fundamental open questions in this area. Such fundamental questions include characterizing the homology of the space of long knots in Euclidean space as a Poisson algebra, as well as giving new constructions and addressing the issue of completeness in the theory of finite-type knot invariants.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A good introduction to Vassiliev's singularity theory approach.

A more recent survey.

Computations by Turchin arising from this approach.

A reference for the de Rham theory approach to knot spaces.

Notes by Longoni explaining the connection to quantum field theory.

An approach through three-manifold topology by Budney and Hatcher.

An introduction to the connection between knot spaces and calculus of the embedding functor by D. Sinha.