Contact Topology, Knots, and Heegaard Floer Theory

Abstract Award: DMS-0805836 Principal Investigator: Olga Plamenevskaya This proposal focuses on contact topology in dimension 3 and the related version of knot theory, the study of Legendrian and transverse knots. Contact structures on 3-manifolds encode subtle topological information, and knots in contact manifolds (tangent or transverse to the contact planes) have many additional properties. The goal of this project is to better understand the relation between and find new applications of various invariants of contact structures and Legendrian and transverse knots. More specifically, the PI will study various versions of the contact, Legendrian and transverse invariants in Heegaard Floer homology; she proposes to investigate knots via associated contact manifolds, such as those obtained by surgery on Legendrian knots and branched covers of transverse knots. A related part of the project is to develop similar invariants in knot homologies arising from quantum algebra (Khovanov and Khovanov-Rozansky homology). An intriguing relation between Heegaard Floer theory (defined via holomorphic disks) and the quantum algebraic knot homologies exists outside of contact geometry; the PI proposes to study this relation in presence of a contact structure. In particular, in her previous work the PI suggested a transverse knot invariant in Khovanov homology. The interplay between Heegaard Floer and Khovanov homologies allows to use this invariant to prove tightness of certain contact structures on branched double covers of knots. Further progress on this project could yield quantum algebraic tools for studying contact structures in a more general setting, and would improve our understanding of the relation between theories of a very different nature. The proposed research is on geometric topology, an area of mathematics that studies shapes of curved spaces (manifolds) in various dimensions. In dimensions 3 and 4, this can be thought of as understanding the structure of space and space-time, and is particularly interesting and important. The study of knots in dimension 3 plays a major role in topology and has important connection to other sciences (for example, DNA and certain proteins can be knotted). The proposed project focuses on study of 3-manifolds and knots in presence of a contact structure, an object somewhat analogous to an electric field in physics. (Historically, the study of contact structures was first motivated by classical mechanics, optics and thermodynamics.) Contact structures are important objects by themselves, but also encode valuable information about the space they live in. In her research, the PI plans to use tools and ideas from different branches of mathematics such as geometry and quantum algebra. The broader goals of the project thus include a better understanding of the relation between these different branches, as well as new results in and applications of the topology of contact manifolds.