  We give the first known topological model for the HOMFLY-PT polynomial. More
precisely, we prove that this invariant is given by a set of graded
intersections between explicit Lagrangian submanifolds in a fixed configuration
space on a Heegaard surface for the link exterior. The submanifolds are
supported on arcs and ovals on the surface.
  The construction also leads to a topological model for the Jones polynomial
constructed from Heegaard surfaces associated directly to the link diagram. In
particular, it does not rely on a choice of a braid representative for the
link. This opens up new avenues for investigation of the geometry of these
invariants, as well as categorifications of geometric nature.
