Narrow Results

We introduce topological helicity, an invariant for oriented framed links. Topological helicity provides an elementary means of computing helicity for a magnetic flux rope by measuring its knotting, linking, and twisting. We present an equivalence relation, reconnection-equivalence, for framed links and prove that topological helicity is a complete invariant for the resulting equivalence classes. We conclude by showing that one can use magnetic reconnection to transform one collection of linked flux ropes into another collection if and only if they have the same helicity.

Recently Nieto has proposed a link between oriented matroid theory and the Schild type action of p-branes. This particular matroid theory satisfies the local condition, i.e., the degenerate form must be closed. This allows us to explain the dynamics of p-branes in terms of Nambu–Poisson structure. In this paper using an infinitesimal canonical transformation of Nambu brackets we show that the helicity is conserved in the dynamics of p-branes. Applying Filippov algebra (or quantum Nambu bracket) we define a generalized Yang–Mills action in 4k space. We show that this action is equivalent to Dolan–Tchrakian type action.