The proposed project is devoted to deepen the connection of two seemingly different subfields of low dimensional topology, namely contact topology and Heegaard Floer homologies. The invariants of Legendrian knots in Heegaard Floer homology gave several first proofs of new phenomenons in knot theory. During this project the PI would like to understand relations between the already defined knot invariants for Legendrian knots, see how these invariants are related for concordant Legendrian knots and use these invariants together with other tools to distinguish and hopefully classify Legendrian representations of a wide class of knots. Contact geometry provided a tool for proving results in Heegaard Floer homology. As part of the proposed projects the PI would like to use tools from contact geometry to give a lower bound for the rank of Heegaard Floer homology for a manifold that contains an incompressible surface.
Low dimensional topology describes our physical world; the space we live in is a 3-dimensional manifold while space-time is a 4-dimensional manifold. The proposed project is aimed to deepen the connection between two subfields of low dimensional topology that are inspired by physics in different ways. The first is the theory of contact manifolds, which arise quite naturally as phase spaces of moving objects. The other is Heegaard Floer homology which is obtained by associating to an object a configuration space of constrained functions and computing the space's algebro-topological invariants.
