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.
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Operads and knot spaces.
Journal of the American Mathematical Society, Vol 19 No 2 (2006) 461-486.
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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.
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.
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The topology of spaces of knots.
Submitted.
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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.
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A one-dimensional embedding complex. (with
Kevin Scannell)
Journal of Pure and Applied Algebra 170 (2002) 93-107
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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.
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