Homotopy, link homotopy and (higher?) helicity
Seminar Room 1, Newton Institute
I will outline a program for relating link homotopy classes of links in Euclidean 3-space to homotopy classes of certain associated maps. This seems worthwhile since interpreting the linking number - which is the simplest link homotopy invariant - as a homotopy invariant leads directly to the famous Gauss linking integral. Indeed, this approach has already yielded a generalized Gauss integral for Milnor's triple linking number.
The overarching goal is to find invariants of vector fields which will be relevant in, e.g., plasma physics. Specifically, these invariants should be "higher" analogues of helicity, meaning invariants of vector fields which are preserved under volume-preserving diffeomorphisms isotopic to the identity and which provide lower bounds for the field energy. Since helicity can be interpreted as an asymptotic version of the Gauss linking integral, the hope is that higher helicities can be defined as asymptotic versions of generalized Gauss integrals for higher linking invariants.