Geodesic
On the complexity of braids
Journal of the European Mathematical Society, Tome 9 (2007) no. 4, pp. 801-840.

Voir la notice de l'article provenant de la source EMS Press

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​, which are Garside-like half-twists involving strings i through j, and by counting powered generators Δijk​ 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.
DOI : 10.4171/jems/98
Classification : 20-XX, 00-XX
Keywords: Braid, curve diagram, complexity, lamination, train track 
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Ivan Dynnikov; Bert Wiest. On the complexity of braids. Journal of the European Mathematical Society, Tome 9 (2007) no. 4, pp. 801-840. doi : 10.4171/jems/98. http://geodesic.mathdoc.fr/articles/10.4171/jems/98/

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