A topological model for the HOMFLY-PT polynomial

Abstract

We give the first known topological model for the HOMFLY-PT polynomial constructed directly from link diagrams. More precisely, we prove that this invariant is given by graded intersections between explicit Lagrangian submanifolds in a fixed configuration space on a Heegaard surface for the link exterior. The submanifolds are supported on a collection of arcs and ovals on the Heegaard surface. We also obtain two topological models for the Jones polynomial via a Heegaard surface associated to a link diagram. This opens up new avenues for constructing categorifications for the Jones and the HOMFLY-PT polynomials of a geometric nature.