| Heegaard Floer homology is an extremely powerful invariant for closed oriented three-manifolds, introduced by Peter Ozsvath and Zoltan Szabo. This invariant was later generalized by them and independently by Jacob Rasmussen to an invariant for knots inside three-manifolds called knot Floer homology, which was later even further generalized to include the case of links. However the boundary maps in the Heegaard Floer chain complexes were defined by counting the number of points in certain moduli spaces, and there was no algorithm to compute the invariants in general.;The primary aim of this thesis is to address this concern. We begin by surveying various areas of this theory and providing the background material to familiarize the reader with the Heegaard Floer homology world. We then describe the algorithm which was discovered by Jiajun Wang and me, that computes the hat version of the three-manifold invariant with coefficients in F 2. For the remainder of the thesis, we concentrate on the case of knots and links inside the three-sphere. Based on a grid diagram for a knot and following a paper by Ciprian Manolescu, Peter Ozsvath and me, we give a another algorithm for computing the knot Floer homology. We conclude by generalizing the construction to a theory of knot Floer homotopy. |
