On the Ozsvath-Szabo invariant of negative definite plumbed 3-manifolds

Abstract

The main goal of the present article is the computation of the Heegaard Floer homology introduced by Ozsváth and Szabó for a family of plumbed rational homology 3-spheres. The main motivation is the study of the Seiberg-Witten type invariants of links of normal surface singularities.

Knot Floer homology and rational surgeries

“…The set of L-spaces includes all lens spaces and indeed all three-manifolds with elliptic geometry (cf our paper [24]); for more examples; see also Némethi [13]. Another interesting family is given by the branched double-covers of alternating knots in S 3 ; cf our paper [25].…”

“…We describe here a natural F OEU -module associated to h. It is interesting to compare the following construction with a construction of Némethi [13]; see also [17].…”

Links and analytic invariants of superisolated singularities

“…Here, by definition, sw(M ) is the sign-refined Reidemeister-Turaev torsion T (M ) (associated with the canonical spin c structure) [38] normalized by the Casson-Walker invariant λ(M ) (using the convention of [15]) (cf. also with [26,27,28,23,25]). Namely, we consider:…”

“…for quotient singularities [26], for singularities with good C * -actions [27], hypersurface suspension singularities g(u, v)+w n with g irreducible [28]. Even more, in [23], the third author replaced sw(M ) by the corresponding Ozsváth-Szabó invariant (which is defined via the Ozsváth-Szabó Floer homology, and which conjecturally equals sw(M )), and verified the inequality (a) for any singularity with almost rational (AR in short) resolution graph. (A graph is AR, if by replacing the decoration of one of the vertices one gets a rational graph.…”

Heegaard Diagrams and Holomorphic Disks

“…According to [81], L-spaces admit no (coorientable) taut foliations. A striking theorem of Némethi [69] characterizes those L-spaces which are boundaries of negative-definite plumbings of spheres: they are the links of rational surface singularities. Note also that there is an analogous class of three-manifolds in the context of Seiberg-Witten monopole Floer homology, c.f.…”