Combinatorial aspects of Heegaard Floer homology for knots and links

The action's goal is to achieve major advances in Heegaard Floer homology for knots and links. Heegaard Floer homology is a package of powerful invariants for 3-manifolds, and knots and links inside them. Introduced two decades ago, it is now a major research area in low-dimensional topology.

To a knot or link in the 3-sphere, together with extra data called `decoration', Heegaard Floer homology associates a bigraded vector space which determines key topological properties of such a knot or link, such as its Alexander polynomial and its Seifert genus.

Moreover, given a (decorated) link cobordism between two links, there is a linear map induced between their Heegaard Floer homology.

The original definition of Heegaard Floer homology is based on counting pseudo-holomorphic curves in symplectic manifolds, but there exist combinatorial reformulations of the vector spaces associated to decorated knots and links.The proposal consists of three major projects:1) Give a combinatorial reformulation of the Heegaard Floer cobordism maps, to make their computation algorithmic, by extending existing combinatorial definitions of the vector spaces associated to decorated knots and links.2) Extend the most efficient combinatorial reformulation, namely the Kauffman-states functor, from decorated knots to decorated links.3) Define a combinatorial Heegaard Floer invariant for partially decorated links, for which attempts to give an analytic definition seems unfeasible.