Nonseparating spheres and twistedHeegaard Floer homology

Heegaard Floer homology was introduced by Ozsváth and Szabó [16]. For nullhomologous knots, there is a filtered version of Heegaard Floer homology, called knot Floer homology; see Ozsváth and Szabó [14] and Rasmussen [18]. Basically, if one knows the information about the knot Floer homology of a knot, then one can compute the Heegaard Floer homology of any manifold obtained by Dehn surgery on the knot. However, in general the algebra involved here is too complicated. In order to get useful information, people often assume the ambient manifold has “simple” Heegaard Floer homology, namely, the ambient manifold is an L–space.