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 Teichmueller 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 r\^ole in the proofs is played by a technique introduced by Agol, Hass, and Thurston.Comment: Version 2: added section on Teichmueller geometry, removed section on train track