The main goal of this project is the study of classical knots and more general spaces of embeddings through a relatively new theory of calculus of the embedding functor, developed by T. Goodwillie and M. Weiss. Recent results of Volic show that there is a strong connection between the two. In fact, a certain tower of spaces arising from calculus of functors serves as a classifying object for finite type knot invariants, a fascinating class of invariants which has been found to connect in intricate ways to other areas of topology and geometry, as well as physics. Other goals of the project concern more general spaces of knots and in particular the collapse of certain spectral sequences associated to these spaces. This should give new information about homology and homotopy of spaces of knots, as well as new insight into appearance of interesting combinatorics in their study. A recent result due to P. Lambrechts and Volic concerning configuration spaces will serve as the starting point for this part of the project. Since the construction of the calculus tower modeling the space of knots is quite general, another goal is to extract information about other spaces of embeddings. In particular, G. Arone and Volic plan to use the interplay of two versions of calculus of the embedding functor, manifold and orthogonal, to show that the spectral sequences arising from these theories collapse. This should also result in new insight into how these two versions of calculus interact.
Knots are some of the most interesting objects of study in topology both because they are easy to define and visualize and because they are of interest to physicists, chemists, etc. Some fundamental questions about knots, such as their classification, or construction of efficient ways of telling them apart (i.e. finding good knot invariants), still generate a wealth of exciting research. One of the main objectives of this project is to further the understanding of knot theory by studying it through the new technique of calculus of functors. It turns out, however, that the methods used are quite general and extend beyond knots to larger classes of topological spaces. Thus the new connections between topology, geometry, combinatorics, and physics which are expected to arise from this project could have broad implications as well as bring together various schools of thought in topology in unexpected ways.
