  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].
