We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures. In this article we explain how to normalize tight contact structures with respect to a fixed triangulation. Using t...

In this paper we first give a one-move version of Markov’s braid theorem for knot isotopy in S that sharpens the classical theorem. Then a relative version of Markov’s theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markov’s theorem for knot isotopy in knot complements. Finally we extend this last result to prove a Markov theorem for links in an arbitrary or...

In this article we classify up to isotopy the tight contact structures on Seifert manifolds over the torus with one singular fibre.