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.