Ozsváth-Szabó invariants and tight contact 3-manifolds, III

Abstract

We characterize L -spaces which are Seifert fibered over the 2-sphere in terms of taut foliations, transverse foliations and transverse contact structures. We give a sufficient condition for certain contact Seifert fibered 3-manifolds with e 0 = −1 to have nonzero contact Ozsváth-Szabó invariants. This yields an algorithm for deciding whether a given small Seifert fibered L -space carries a contact structure with nonvanishing contact Ozsváth-Szabó invariant. As an application, we prove the existence of tight contact structures on some 3-manifolds obtained by integral surgery along a positive torus knot in the 3-sphere. Finally, we prove planarity of every contact structure on small Seifert fibered L -spaces with e 0 ≥ −1 , and we discuss some consequences.

Heegard Floer invariants of Legendrian knots in contact three-manifolds

“…now [17,Section 5]). In contrast, for the non-loose knots L(n) of the previous subsection (also having non-trivial L-invariants) the same intuitive argument does not work, since some negative surgery on the knot L(n) will produce a contact structure on the 3-manifold S 3 2n−1 (T (2,2n+1) ) and since this 3-manifold does not admit any tight contact structure [24], the result of the surgery will be overtwisted independently of the chosen stabilizations. Nevertheless, the overtwisted disk in the 3-manifold obtained by only negative stabilizations cannot be in the complement of the knot L(n), since such stabilizations are still non-loose (shown by the non-vanishing of the invariant L(L(n))).…”

“…As is explained in [25,Section 6], the Legendrian link underlying the surgery diagram for ξ n (together with the Legendrian knot L(n)) can be put on a page of an open book decomposition with planar pages, which is compatible with the standard contact structure ξ st on S 3 . This can be seen by considering the annular open book decomposition containing the Legendrian unknot (and its Legendrian push-offs), and then applying the stabilization method described in [11] for the stabilized knots.…”

Cable links and L-space surgeries

“…Then ℓ consists of only d * i segments, i ∈ Z. It is straightforward to see (for example, by considering the segments as drawn in [HW15, Figure 1 of [LS07], we see that M (−1; 1/p 1 , . .…”

A combinatorial description of the Heegaard Floer contact invariant

“…Y / (defined up to sign for the theory with ‫ޚ‬ coefficients). Since non-vanishing of c. / implies that is tight, this invariant gives a powerful tool for establishing the tightness of a contact structure (see Lisca and Stipsicz [5;6;7]). …”