Train tracks and zipping sequences for pseudo-Anosov braids

The n-string braid group is naturally isomorphic to the group of isotopy classes of orientation-preserving diffeomorphisms from the n-punctured closed disk to itself which restrict to the identity map on the boundary of the disk. Every such diffeomorphism may be classified as either periodic, reducible, or pseudo-Anosov. In the pseudo-Anosov case, the map may be described in terms of an invariant train track. In his dissertation, H. Huang defined the dynamical power of the center, a number which describes how the diffeomorphism behaves outside a regular neighborhood of the invariant train track. Another object, introduced by Papadopoulos and Penner in 1987 under the name “RLS word”, describes how the map behaves inside a regular neighborhood of the train track. We introduce the concept of zipping sequences, which encode the same information as RLS words in a more useful and easily visualized manner, and we sharpen the definition of zipping sequences by proving that the zipping can always be done from just one outside cusp of the train track. We prove that the ordered triple consisting of the invariant train track, the dynamical power of the center, and the zipping sequence completely determines a braid up to conjugacy. As an application, we present a new solution of the conjugacy problem for pseudo-Anosov 4-braids. In addition, we describe an algorithm for computing the dynamical power of the center of a pseudo-Anosov 4-braid. We also indicate how zipping sequences are related to criteria for recognizing when a closed braid is Artin reducible.