Ozsváth–Szábo invariants and tight contact three-manifolds I

Abstract

Let S 3 r (K) be the oriented 3-manifold obtained by rational r-surgery on a knot K ⊂ S 3 . Using the contact Ozsváth-Szabó invariants we prove, for a class of knots K containing all the algebraic knots, that S 3 r (K) carries positive, tight contact structures for every r = 2g s (K) − 1, where g s (K) is the slice genus of K . This implies, in particular, that the Brieskorn spheres −Σ(2, 3, 4) and −Σ(2, 3, 3) carry tight, positive contact structures. As an application of our main result we show that for each m ∈ N there exists a Seifert fibered rational homology 3-sphere M m carrying at least m pairwise non-isomorphic tight, nonfillable contact structures.

On overtwisted, right-veering open books

“…We refer the reader to [Ding and Geiges 2004] for this fact and the basic properties of contact surgeries, and to [Lisca and Stipsicz 2004] for the use of the "front notation" in contact surgery presentations, in particular for the meaning of Figure 2 below. Proposition 2.1.…”

Tight Contact Structures on Laminar Free Hyperbolic Three-Manifolds

“…In fact, it can be obtained by (−m + 2)-surgery on the right-handed trefoil as demonstrated in Figure 5. Since the maximal Thurston-Bennequin number and the 4-ball genus of the right-handed trefoil are both 1, the main result of [15] implies that M (m; 1/1) admits a tight contact structure.…”

“…On the other hand, the Legendrian knot L with contact framing r in Figure 4 has ThurstonBennequin invariant tb(L) = 1, its smooth type is (the positive) 5 2 knot K which has 4-ball genus g 4 (K) = 1. It is known that if a Legendrian knot in S 3 with the standard contact structure satisfies tb(L) = 2g 4 (K) − 1, then any contact surgery along L with positive contact framing results in a contact structure with nonzero contact invariant [15]. Contact surgery with r = 0 framing is not well-defined since all the contact structures on the filling solid torus which can be glued to the complement of a standard neighborhood of the surgery curve are overtwisted.…”

Notes on the contact Ozsváth–Szabó invariants

“…If K is smoothly isotopic to a negative torus knot, then (Y K , ξ K ) is overtwisted. [ Lisca and Stipsicz 2004a]. The tightness question for contact structures can be fruitfully attacked with the use of the contact Ozsváth-Szabó invariants [Ozsváth and Szabó 2005].…”

“…The tightness question for contact structures can be fruitfully attacked with the use of the contact Ozsváth-Szabó invariants [Ozsváth and Szabó 2005]. In fact, the nonvanishing of these invariants implies tightness, while their computation can sometimes be performed (see, e.g., [Lisca and Stipsicz 2004a;) using a contact surgery presentation in conjunction with the surgery exact triangle established in Heegaard Floer theory by Peter Ozsváth and Zoltán Szabó [2003a]. Such ideas can be used to prove the next theorem.…”