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We continue our study of the monoid of right-veering diffeomorphisms on acompact oriented surface with nonempty boundary, introduced in [HKM2]. Weconduct a detailed study of the case when the surface is a punctured torus; inparticular, we exhibit the difference between the monoid of right-veeringdiffeomorphisms and the monoid of products of positive Dehn twists, with thehelp of the Rademacher function. We then generalize to the braid group B_n on nstrands by relating the signature and the Maslov index. Finally, we discuss thesymplectic fillability in the pseudo-Anosov case by comparing with the work ofRoberts [Ro1,Ro2].Comment: 25 pages, 5 figure

Right-veering diffeomorphisms of compact surfaces with boundary