Can complex mathematical structures be simplified and classified? This paper explores this question within the realm of braid theory, a branch of mathematics dealing with the properties of interwoven strands. The authors classify pseudo-Anosov homeomorphisms, transformations that stretch and fold surfaces in a chaotic manner, using their invariant train tracks. They develop a normal form for each train track class by decomposing train track maps into elementary folding maps. Using **mathematics**, they give an explicit automaton that generates a normal form for each class. Also, for instance, to exhibit the pseudo-Anosov 4-braid with the minimal growth rate, which helps to understand braid structure. This simplifies the analysis of 4-braids. They demonstrate that the growth rate of a pseudo-Anosov braid is a root of the Alexander polynomial of a related link and provide a criterion for the faithfulness of the Burau representation. The results advance our understanding of braid groups and their connections to other mathematical fields such as knot theory and dynamical systems. Future work can explore generalizations of these findings to more complex braid groups and their applications in physics and computer science.
Published in the _Journal of Knot Theory and Its Ramifications_, this paper is aligned with the journal’s aims of advancing the field of knot theory and related areas. By examining the classification and properties of braids, it supports the journal’s goal of exploring complex topologies and their mathematical implications.