Combinatorial Knot Floer Homology

Abstract

Knot Floer Homology HFK(K) of a knot K in S^3 was defined by Peter Ozsváth and Zoltan Szabó in 2003. It has since become powerful invariant for the study of properties of knots. The definition of the Knot Floer Homology groups initially involved counting holomorphic disks in symmetric products of Riemann surfaces, which made them difficult to compute. In 2006, in a paper titled "A combinatorial Description of Knot Floer Homology", the authors Ciprian Manolescu, Peter Ozsváth, and Sucharit Sarkar discovered an algorithm for purely combinatorial description of knot Floer homology, making its computation, in principle, fully accessible. This thesis defense will describe their combinatorial algorithm along with examples and applications.