Floer homology
Several kinds of Floer homology are special cases of Lagrangian Floer homology. The symplectic Floer homology of a symplectomorphism of M can be thought of as a case of Lagrangian Floer homology in which the ambient manifold is M crossed with M and the Lagrangian submanifolds are the diagonal and the graph of the symplectomorphism. The construction of Heegaard Floer homology is based on a variant of Lagrangian Floer homology for totally real submanifolds defined using a Heegaard splitting of a three-manifold. Seidel-Smith and Manolescu constructed a link invariant as a certain case of Lagrangian Floer homology, which conjecturally agrees with Khovanov homology, a combinatorially-defined link invariant.