Tight contact structures on some small Seifert fibered 3-manifolds

Abstract

We classify tight contact structures on the small Seifert fibered 3-manifold M (−1; r 1 , r 2 , r 3 ) with r i ∈ (0, 1) ∩ Q and r 1 , r 2 ≥ 1 2 . The result is obtained by combining convex surface theory with computations of contact Ozsváth-Szabó invariants. We also show that some of the tight contact structures on the manifolds considered are nonfillable, justifying the use of Heegaard Floer theory.

Heegard Floer invariants of Legendrian knots in contact three-manifolds

“…In fact, by stabilizing L k,l on the left and then performing a contact (−1)-surgery we still get a tight contact 3-manifold: it will be Stein fillable if we perform only one stabilization, and not Stein fillable but tight for more stabilizations. The tightness of the result of these latter surgeries was verified in [17] by computing the contact Ozsváth-Szabó invariants of the resulting contact structures. Notice that this observation implies that after arbitrarily many left stabilizations L k,l remains non-loose, which, in view of Proposition 5.2, is a necessary condition for L(L k,l ) to be non-vanishing.…”

“…(Finally, note that performing contact (−1)-surgery on L k,l after a single right stabilization provides an overtwisted contact structure; cf. 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.…”

Planar open books, monodromy factorizations and symplectic fillings

“…(We use this notation for the space given by the surgery diagram of Figure 1; here and throughout the paper, r 1 ; r 2 ; r 3 are rational numbers between 0 and 1.) Tight contact structures on such manifolds were studied by Ghiggini-LiscaStipsicz [8] and 20]; when r 1 ; r 2 1=2, a complete classification of tight contact structures on M. 1I r 1 ; r 2 ; r 3 / was obtained in [8] (in particular, each of these spaces is known to carry a tight contact structure). Tightness of some of these contact structures was established by means of the Heegaard Floer theory; it was shown in [8] that one of the tight structures on M. 1I 1=2; 1=2; 1=p/ is nonfillable.…”

“…Tight contact structures on such manifolds were studied by Ghiggini-LiscaStipsicz [8] and 20]; when r 1 ; r 2 1=2, a complete classification of tight contact structures on M. 1I r 1 ; r 2 ; r 3 / was obtained in [8] (in particular, each of these spaces is known to carry a tight contact structure). Tightness of some of these contact structures was established by means of the Heegaard Floer theory; it was shown in [8] that one of the tight structures on M. 1I 1=2; 1=2; 1=p/ is nonfillable. (Recall that, in contrast, all tight contact structures on M.0I r 1 ; r 2 ; r 3 / are fillable; see Ghiggini-Lisca-Stipsicz [7], cf Wu [28]).…”

On tight contact structures with negative maximal twisting number on small Seifert manifolds

“…The generic case, when e 0 ¤ 1; 2, was settled by Wu [20] and the author, Lisca and Stipsicz [5], who also studied a large family of manifolds with e 0 D 1 in [6]. The goal of this article is the classification of tight contact structures on some small Seifert manifolds with e 0 D 2.…”