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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.

PS PDF Operads and knot spaces.
Journal of the American Mathematical Society, Vol 19 No 2 (2006) 461-486.
In this paper I show that the totalization of the Kontsevich operad gives a model for certain knot spaces, which on passing to spectral sequences gives results close to those of Tourtchine and establishes a corrected version of a conjecture of Kontsevich from his New Directions talk at UCLA in 2000. The material on the Kontsevich operad and a new operad called the choose-two operad may be of independent interest.

PS PDF New perspectives on self linking (with Ryan Budney, Kevin Scannell, and James Conant)
Advances in Mathematics, Vol 191 No 1 (2005), 78-113.
In this paper we give a new interpretation of the simplest finite-type invariant of knots in terms of counting collinearities of four points on the knot. We show that this invariant is the only knot invariant which arises in the Goodwillie tower for classical knots through degree three. What unifies these results is the evaluation map of a knot, and cobordism invariants thereof. Extending these restults to higher degrees is one of my main projects.

PS PDF The topology of spaces of knots.
This paper is the foundation on which I build in my other papers on this subject. I produce two models equivalent to the Goodwillie-Weiss models for knot spaces, each with different advantages. The first model makes clear the central role of the evaluation map, from configurations on the knot to configurations in the ambient space, which can be used for further developments in differential topology. The second model is cosimplicial, giving efficient means for calculations for homotopy and cohomology groups.

PS PDF A one-dimensional embedding complex. (with Kevin Scannell)
Journal of Pure and Applied Algebra 170 (2002) 93-107
In this paper we compute the E1 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 E1 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 &pii of the space of knots in R4 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.