A topological model for the HOMFLY-PT polynomial

TL;DR

This paper introduces a topological model for the HOMFLY-PT polynomial based on graded intersections of Lagrangian submanifolds in a Heegaard surface, providing a geometric approach to link invariants.

Contribution

It presents the first direct topological model for the HOMFLY-PT polynomial from link diagrams, linking it to Lagrangian intersections in a Heegaard surface.

Findings

Established a topological model for the HOMFLY-PT polynomial.

Derived models for the Jones polynomial using Heegaard surfaces.

Opened new pathways for geometric categorifications of link polynomials.

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.