Knot Floer homology and rational surgeries

Abstract

Let K be a rationally null-homologous knot in a three-manifold Y . 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 K . As an application, we express the Heegaard Floer homology of rational surgeries on Y along a null-homologous knot K in terms of the filtered homotopy type of the knot invariant for K . This has applications to Dehn surgery problems for knots in S 3 . 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.

An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold

“…Section 4 of [30], describes the relationship between the knot filtration and the Ozsváth-Szabó Floer homologies of three-manifolds obtained by performing "sufficiently large" integral surgeries on Y along K. Moreover, this relationship gives an interpretation of some of the maps induced by cobordisms in terms of the knot filtration. These results were generalized to include all rational surgeries on knots in rational homology spheres in [34,35], but the results of [30] will be sufficient for our purposes. We review these results here, and refer the reader to [30,34,35] for a more thorough treatment.…”

Non-left-orderable surgeries and generalized Baumslag–Solitar relators

“…For instance, any given knot in the 3-sphere S 3 admitting an L-space by surgery gives rise to an infinite family of L-spaces. In fact, it was shown in [8,Proposition 9.6] that if a knot K has one L-space surgery, then all the surgered manifolds are L-spaces if the surgery slopes are greater than or equal to 2g(K) − 1, where g(K) denotes the genus of the knot. Precisely, for a given knot K in a 3-manifold M , the following operation is called Dehn surgery; removing an open regular neighborhood of K from M , and gluing a solid torus back.…”

On Floer homology and the Berge conjecture on knots admitting lens space surgeries

“…To do so recall that to any knot (Y, K) in a rational homology sphere (i.e. H * (Y ; Q) ∼ = H * (S 3 ; Q)), Ozsváth and Szabó associate a collection of bigraded groups (see [27]), …”