Heegaard floer homolojisi

P. Ozsvath and Z. Szabo recently introduced Heegaard Floer homology, an invariant forclosed oriented 3-manifolds associating to each such manifold a sequence of finitely generatedabelian groups. The construction has also been extended to an invariant for knots, and thenfor links.The definition of Heegaard Floer homology involves many steps, such as Heegaard decompositionsand pointed Heegaard diagrams of 3-manifolds, symmetric products of surfacesand counting some holomorphic representatives of disks on the symmetric product. Certainvariations concerning these steps lead to four different types of homologies.With additional basepoints in the Heegaard diagram, one can obtain knots or links ina 3-manifold and consequently the Heegaard Floer homologies become invariants for knotsand links. They are called knot (or link) Floer homologies. In this thesis, only knots andlinks in the 3-sphere are studied, but it should be noted that the ideas are applicable forknots and links in an arbitrary closed oriented 3-manifold.In the final chapter, we analyze a combinatorial way of computing knot and link Floerhomologies, due to C. Manolescu, P. Ozsvath, and S. Sarkar. The idea is to use somespecial Heegaard diagrams, in order to project the knot to a grid diagram and compute thedifferential map in a purely combinatorial way. We also include computations of knot Floerhomologies for the trefoil and the figure eight knot with the help of the MATLAB software.