On the complexity of braids

Abstract

We define a measure of “complexity” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators \Delta_{ij} , which are Garside-like half-twists involving strings i through j , and by counting powered generators \Delta_{ij}^k as \log(|k|+1) instead of simply |k| . The geometrical complexity is some natural measure of the amount of distortion of the n times punctured disk caused by a homeomorphism. Our main result is that the two notions of complexity are comparable. This gives rise to a new combinatorial model for the Tei space of an n+1 times punctured sphere. We also show how to recover a braid from its curve diagram in polynomial time. The key rôle in the proofs is played by a technique introduced by Agol, Hass, and Thurston.