Homotopy Methods in Knot Theory

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

I propose to use methods from algebraic topology in the study of knots. A knot induces a map from the space of configurations on the knot to configurations in the ambient manifold. With recently developed compactification technology, one can fix boundary conditions on those configuration spaces and study the homotopy class of this induced map relative to those boundary conditions. Such an approach was pioneered by Bott and Taubes in de Rham theory and by the PI and his collaborators in homotopy theory itself. Results of Volic may be used to show that all Bott-Taubes invariants, and thus all real finite-type invariants, are homotopy invariants of this induced map. Already in lowest degree, new geometric understanding arises from studying the induced map directly homotopy, and I propose to continue this study in higher degrees. I also propose to more deeply understand the role of operads in this theory, as well as to extend these techniques to study link homotopy. Knot theory, the study of embedded loops in space, is one of the oldest and most distinguished fields in topology. For most of its life, knot theory has developed in parallel with other subfields of topology. But in the last twenty years, the field has changed dramatically from the influence of previously unrelated fields. In particular, quantum field theory has provided ground-breaking new constructions. One can try to define the energy of a knot as an invariant, but in order to do so one must "integrate over all connections", that is over all ways of putting an "energy field" on the space in which the knot lives. Such an integral does not exist in precise mathematical form, but through the standard perturbative expansion in quantum field theory one can write down Feynman integrals which are meant to approximate it to finite order. Topologists have made such integrals precise and rigorous in this setting, and shown that they provide a basis for the finite-type invariants of knots. But topologists would like to reconnect the theory to more standard constructions in topology. The PI's previous work marks the beginning of such a connection. New geometric insight was gained in the process, as the simplest quantum invariant can now be computed by counting instances of a line intersecting a knot in exactly four places. I hope to find both new connections with classical topology and novel geometric interpretations in this proposed work.