We compute the generalized Lefschetz number of orientation-preserving self-homeomorphisms of a compact punctured disk, using the fact that homotopy classes of these homeomorphisms can be identified with braids. This result is applied to study Nielsen-Thurston canonical homeomorphisms on a punctured disk. We determine, for a certain class of braids, the rotation number of the corresponding canonical homeomorphisms on the outer boundary circle. As a consequence of this result on the rotation number, it is shown that the canonical homeomorphisms corresponding to some braids are pseudo-Anosov with associated foliations having no interior singularities.
Takashi MATSUOKA. "The generalized Lefschetz number of homeomorphisms on punctured disks." J. Math. Soc. Japan 61 (4) 1205 - 1241, October, 2009. https://doi.org/10.2969/jmsj/06141205