Fundamental Groups, 3-Braids, and Effective Estimates of Invariants
We define invariants of braids rather than invariants of conjugacy classes of braids. For any pure three-braid we give effective upper and lower bounds for these invariants. This is done in terms of a natural syllable decomposition of the word representing the image of the braid in the braid group modulo its center. The bounds differ by a multiplicative constant not depending on the word. Respective bounds are given for all three-braids. We also obtain effective upper and lower bounds for the entropy of pure three-braids in these terms. The proof leads to the study of the extremal length of classes of curves representing elements of the fundamental group of the twice punctured complex plane.
math.GT↗