#
Moduli spaces of knots

January 3 to January 7, 2006
at the

American Institute of Mathematics,
San Jose, 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:
- 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.
- 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.
- 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.
- 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.