ABSTRACT. We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary, introduced in [HKM2]. We conduct a detailed study of the case when the surface is a punctured torus; in particular, we exhibit the difference between the monoid of right-veering diffeomorphisms and the monoid of products of positive Dehn twists, with the help of the Rademacher function. We then generalize to the braid group B n on n strands by relating the signature and the Maslov index. Finally, we discuss the symplectic fillability in the pseudo-Anosov case by comparing with the work of Roberts [Ro1,Ro2].

We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhász's sutured Floer homology. Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsváth and Szabó.

ABSTRACT. We present an alternate description of the Ozsváth-Szabó contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M, ξ) has an adapted open book decomposition whose page S is a once-punctured torus, then the monodromy is rightveering if and only if the contact structure is tight.

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