which are asymptotic to invariant solutions at infinity and negative infinity. One version of Seiberg–Witten–Floer homology was constructed rigorously in the monograph Monopoles and Three-manifolds by Peter Kronheimer and Tomasz Mrowka, where it is known as monopole Floer homology. Taubes has shown that it is isomorphic to embedded contact homology. Alternate constructions of SWF for rational homology 3-spheres have been given by and ; they are known to agree. Heegaard Floer homology Heegaard Floer homology is an invariant due to Peter Ozsváth and Zoltán Szabó of a closed 3-manifold equipped with a spinc structure. It is computed using a Heegaard diagram of the space via a construction analogous to Lagrangian Floer homology. announced a proof that Heegaard Floer homology is isomorphic to Seiberg–Witten Floer homology, and announced a proof that the plus-version of Heegaard Floer homology (with reverse orientation) is isomorphic to embedded contact homology. A knot in a three-manifold induces a filtration on the Heegaard Floer homology groups, and the filtered homotopy type is a powerful knot invariant, called knot Floer homology. It categorifies the Alexander polynomial. Knot Floer homology was defined by and independently by . It is known to detect knot genus. Using grid diagrams for the Heegaard splittings, knot Floer homology was given a combinatorial construction by . The Heegaard Floer homology of the double cover of S^3 branched over a knot is related by a spectral sequence to Khovanov homology . The "hat" version of Heegaard Floer homology was described combinatorially by . The "plus" and "minus" versions of Heegaard Floer homology, and the related Ozsváth–Szabó four-manifold invariants, can be described combinatorially as well . Embedded contact homology Embedded contact homology, due to Michael Hutchings, is an invariant of 3-manifolds (with a distinguished second homology class, corresponding to the choice of a spinc structure in Seiberg–Witten Floer homology) isomorphic (by work of Clifford Taubes) to Seiberg–Witten Floer cohomology and consequently (by work announced by and ) to the plus-version of Heegaard Floer homology (with reverse orientation). It may be seen as an extension of Taubes's Gromov invariant, known to be equivalent to the Seiberg–Witten invariant, from closed symplectic 4-manifolds to certain non-compact symplectic 4-manifolds (namely, a contact three-manifold cross R). Its construction is analogous to symplectic field theory, in that it is generated by certain collections of closed Reeb orbits and its differential counts certain holomorphic curves with ends at certain collections of Reeb orbits.