We study open book foliations on surfaces in 3-manifolds and give applications to contact geometry of dimension 3. We prove a braid-theoretic formula for the self-linking number of transverse links, which reveals an unexpected connection with to the Johnson-Morita homomorphism in mapping class group theory. We also give an alternative combinatorial proof of the Bennequin-Eliashberg inequality.

We introduce essential open book foliations by refining open book foliations, and develop technical estimates of the fractional Dehn twist coefficient (FDTC) of monodromies and the FDTC for closed braids, which we introduce as well.As applications, we quantitatively study the 'gap' between overtwisted contact structures and non-right-veering monodromies. We give sufficient conditions for a 3-manifold to be irreducible and atoroidal. We also show that the geometries of a 3manifold and the complement of a closed braid are determined by the Nielsen-Thurston types of the monodromies of their open book decompositions.

We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic transverse knots with the same self-linking number. These pairs of knots include most of the non-transversely simple knots of Birman-Menasco and Ng-Ozsváth-Thurston. arXiv:0712.1557v1 [math.GT]

Abstract. We study b-arc foliation change and exchange move of open book foliations which generalize the corresponding operations in braid foliation theory. We also define a bypass move as an analogue of Honda's bypass attachment operation.As applications, we study how open book foliations change under a stabilization of the open book. We also generalize Birman-Menasco's split/composite braid theorem: We show that closed braid representatives of a split (resp. composite) link in a certain open book can be converted to a split (resp. composite) closed braid by applying exchange moves finitely many times.

It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links.The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type.We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for K and L; then it is also true for the .p; q/-cable of K and for the connect sum of K and L: 57M25; 57M27

We give an alternative proof of a theorem of Honda-Kazez-Matić that every non-right-veering open book supports an overtwisted contact structure. We also study two types of examples that show how overtwisted discs are embedded relative to right-veering open books.

We introduce a notion of "quasi-right-veering" for closed braids, which plays an analogous role to "right-veering" for open books. We show that a transverse link K in a contact 3-manifold (M, ξ) is non-loose if and only if every braid representative of K with respect to every open book decomposition that supports (M, ξ) is quasi-right-veering. We also show that several definitions of "right-veering" closed braids are equivalent.

We first define a Rohlin property for one-parameter automorphism groups of the hyperfinite type Hi factor as an analogue of Kishimoto's definition for one-parameter automorphism groups of unital simple C*-algebras.Secondly we prove equivalence between the Rohlin property and the cohomology vanishing in an appropriate central sequence algebra, which is a variation of Kishimoto's theorem in C*-algebra theory. §1. IntroductionClassification of group actions on von Neumann algebras was dramatically developed by Connes in the course of proving the uniqueness of aperiodic approximately inner automorphisms of a separable McDuff Hi factor up to outer conjugacy, in particular, he gave a proof to the uniqueness of aperiodic automorphisms of the hyperfinite type IIi factor R, which we sometimes call free integer group actions, i.e., Z-actions, up to outer conjugacy [3]. Later Jones and Ocneanu classified actions (including non-free ones) of finite groups and discrete amenable groups on R, respectively, up to cocycle conjugacy [5] [10]. Then our next interest moves into considering classification of continuous group actions on R, especially one-parameter automorphism groups of R, that is, real group actions on R which we shortly call It-actions on R.