International conference
Combinatorial Methods in Physics and Knot Theory

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. The proof this result yields a new, algorithmically very efficient way to calculate a \emph{canonical representative word} for any element of the braid group $B_n$. We also prove that every braid has a $\sigma_1$-consistent representative of linearly bounded length. The key r\^ole in the proofs is played by a technique introduced by Agol, Hass, and Thurston.