# On the complexity of braids

### Ivan Dynnikov

Moscow State University, Russian Federation### Bert Wiest

Université de Rennes I, France

## 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 $Δ_{ij}$, which are Garside-like half-twists involving strings $i$ through $j$, and by counting powered generators $Δ_{ij}$ as $g(∣k∣+1)$ 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.

## Cite this article

Ivan Dynnikov, Bert Wiest, On the complexity of braids. J. Eur. Math. Soc. 9 (2007), no. 4, pp. 801–840

DOI 10.4171/JEMS/98