Knot Invariants and Nonabelian Hodge Theory via the Unipotent Locus
We'll introduce a construction that turns an n-strand braid into a variety - which we call its cycloform - that lives (equivariantly) over the unipotent locus of SLn. When the braid is the identity, the cycloform is the Springer resolution. In general, we can use the cycloform to build a “Steinberg-like variety” whose equivariant Borel-Moore homology admits an Sn-action, and indeed, the action of a larger algebra. The Steinberg-like variety is interesting in two ways: (1) The Khovanov-Rozansky homology of the link closure of the braid can be recovered from the Sn-action. We expect the algebra action to recover a conjugacy invariant of the braid called the horizontal trace. (2) We conjecture that if the braid maps to an n-cycle, then the variety deformation-retracts onto an (Iwahori) affine Springer fiber for SLn, in a way compatible with the Springer action on both sides and reminiscent of nonabelian Hodge theory. This, in turn, implies unexpected geometric properties of the cycloforms. If time permits, we'll explain the proof of (1) and the numerical evidence for (2). Both extend beyond SLn to other semisimple groups.