On the classification of tight contact structures

Abstract

Abstract. Recently, there have been several breakthroughs in the classification of tight contact structures. We give an outline on how to exploit methods developed by Ko Honda and John Etnyre to obtain classification results for specific examples of small Seifert manifolds.

Tight contact structures of certain Seifert fibered 3-manifolds with e0= −1

“…The special case of Theorem 1.1 for k = 2 is proved in [Ghiggini and Schönenberger 2003]. It would also appear to be interesting to classify tight contact structures on the Brieskorn homology 3-spheres − (2, 3, 6k − 1) (k > 2) equipped with orientation opposite to the one as a boundary of the Milnor fiber.…”

“…We prove Theorem 1.1 by essentially employing the techniques developed in [Honda 2000;Etnyre and Honda 2001] and later implemented in [Ghiggini and Schönenberger 2003]. This paper is a result of the author's attempt to understand those techniques.…”

Classification of tight contact structures on small Seifert 3–manifolds with 𝑒₀≥0

“…More precisely, we classify positive tight contact stuctures on M (r 1 , r 2 , r 3 ) assuming e 0 ≥ 0, by using a set of Legendrian surgery diagrams which is slightly different from the one used in [13]. We will assume the reader's familiarity with the standard techniques of contact topology [2,3,8].…”

Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres

“…The proof is presented assuming the reader is familiar with Giroux's convex surface theory [18] and Honda's bypass technology [23]. The argument parallels the one carried out in Ghiggini-Schönenberger [16], and the reader is referred to that work for an excellent exposition.…”