Quantum Heegaard diagrams and knot Floer Homology

Abstract: We construct a unification of the Alexander and Jones polynomials via a new geometric perspective, involving "quantum Heegaard surfaces". Then, we prove that it leads to knot Floer homology and opens avenues for explicit geometric categorifications for the Jones polynomial. This model codifies the difference between the geometry of the Alexander and Jones polynomials via a quantum deformation of the Alexander grading, constructed on the surface. We do so by introducing a new type of diagram, called "quantum Heegaard diagram" (encoding the knot complement) together with a "quantum Alexander grading". This grading is a refinement of the Alexander grading used in knot Floer homology. Then we define a graded Lagrangian intersection, in two variables, between concrete Lagrangian submanifolds arising from the curves of the diagram in the symmetric power of the quantum Heegaard surface. The two-variable intersection unifies and recovers the Jones and Alexander polynomials as two specialisations of coefficients. Moreover, this leads to knot Floer homology on one side and provides an intersection model for the Jones polynomial on the other. The surface and Lagrangian submanifolds are concretely constructed via braid actions on the punctured disc.