JournalsjemsVol. 9 , No. 4DOI 10.4171/jems/98

On the complexity of braids

  • Ivan Dynnikov

    Moscow State University, Russian Federation
  • Bert Wiest

    Université de Rennes I, France
On the complexity of braids cover


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 Δij\Delta_{ij}, which are Garside-like half-twists involving strings ii through jj, and by counting powered generators Δijk\Delta_{ij}^k as log(k+1)\log(|k|+1) instead of simply k|k|. The geometrical complexity is some natural measure of the amount of distortion of the nn 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+1n+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.