Heegaard Floer homology is a collection of invariants of knots, 3-manifolds, and smooth 4-manifolds, defined using symplectic geometry. It has changed the landscape of low-dimensional topology over the last decade, and has, for example, been instrumental in many of the recent advances in contact geometry. This project is devoted to better understanding the internal workings of this theory and its relationships to other link homology theories like Khovanov and RozanskyÂ¢s categorification of the HOMFLY polynomial. One of the PIÂ¢s broad goals is to use these relationships to develop more computable invariants of knots, 3-manifolds, and contact structures. Another goal involves using Heegaard Floer homology to better understand the correspondence between topological features of open books and geometric properties of contact structures. Specific aims include finding obstructions to a contact structure having support genus one, determining whether Legendrian surgery preserves tightness, understanding the relationship between fractional Dehn twist coefficient and fillability, and developing a combinatorial spanning tree model for knot Floer homology. The PI plans to use the latter to find an axiomatic description of knot Floer homology, in part to show that it agrees with the monopole and instanton knot homologies defined by Kronheimer and Mrowka using gauge theory.
The proposed project involves studying geometric structures on 3- and 4-dimensional manifolds. Understanding these spaces and structures is central to understanding the shape of our macroscopic universe, and some of the theories the PI intends to study have applications and origins in physics. Contact structures, in particular, can serve as a probe with which to study these spaces, and are also important in classical mechanics, thermodynamics, dynamical systems, and in the study of liquid crystals. Another goal of the PIÂ¢s proposal is to develop new methods for understanding knots. Knots arise naturally in molecules like DNA. Recent work, for example, has applied Floer homology to determine how various enzymes in the body alter the knottedness of DNA. Understanding these mechanisms is important for developing certain drugs.