Let be a rationally null-homologous knot in a three-manifold . We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot . As an application, we express the Heegaard Floer homology of rational surgeries on along a null-homologous knot in terms of the filtered homotopy type of the knot invariant for . This has applications to Dehn surgery problems for knots in . In a different direction, we use the techniques developed here to calculate the Heegaard Floer homology of an arbitrary Seifert fibered three-manifold with even first Betti number.
"Knot Floer homology and rational surgeries." Algebr. Geom. Topol. 11 (1) 1 - 68, 2011. https://doi.org/10.2140/agt.2011.11.1