Vassiliev's pioneering work established the study of moduli spaces of knots
as both a useful tool in classical knot theory and an interesting subject in
its own right. One of my goals in the subject is to understand the homotopy
groups and cohomology ring of spaces of knots. A second goal is to provide
a differential-topological framework for the theory
of finite-type invariants, currently grounded in the Kontsevich integral
and thus only fully developed in characteristic zero.
The main tools I apply are the calculus of isotopy functors, as developed by
Goodwillie, Klein and Weiss, and compactification technology as
developed by Fulton, MacPherson and many others including myself
(see my page on configuration spaces).
I am also developing a general theory of Hopf invariants,
in the context of Lie coalgebras and
homotopy theory, which
I plan to use in this setting to define knot invariants.
I describe my research plans further below.
^{1} term of a spectral sequence
converging to the rational homotopy groups of the space of long
knots in an even-dimensional Euclidean space. The complexes given
by rows in this E^{1} term coincide with complexes defined
by Kontsevich in a different context in his New Directions talk at
UCLA in 2000. We prove a vanishing result for Euler
characteristic and make low-dimensional
calculuations, enough for example to show that &pi_{i} of the
space of knots in R^{4} is non-trivial for i=2,3,4,5,6.
These are the first published computations of homotopy groups of knot spaces.
Much more is known about knot spaces than when I started this project, not only from my work described above, but work of others as well. Budney has completely characterized the homotopy types of spaces of knots in Euclidean space of dimension three. Lambrechts, Tourtchine, and Volic determine the rational homology of spaces of knots in higher-dimensional Euclidean spaces. Salvatore and Sakai (in separate work) answer questions I posed about little two-cubes actions on these spaces. As for future work, Budney, Conant, Scannell and I have a conjecture as to the form of a degree-three knot invariant defined using intersections of a knot with circles and parabolas, which would generalize our work on quadrisecants. Conant and I have shown that all knot invariants obtained from the embedding calculus are of finite type, and we are presently writing up that result. Finally, Conant, Longoni, Tourtchine and I have a conjectured form for canonical chains in knot spaces, close in spirit to Vassiliev's singularity theory approach but showing directly why the Poisson operad comes into play. Resolving this conjecture would give a concrete understanding of the homology of knot spaces, which while difficult to compute would be straightforward to visualize. |