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arXiv: math.GT/0403177

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Journal of the European Mathematical Society 9, 4 (2007) 801-840

Available versions:

v1 (2004-03-10)

v2 (2006-04-15)

On the complexity of braids

Ivan Dynnikov 1, Bert Wiest 2

(2007)

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 Teichmueller 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.

1:	Dept. of Mechanics and Mathematics (Dept. of Mechanics and Mathematics, Moscow State University)
	Lomonosov Moscow State University
2:	Institut de Recherche Mathématique de Rennes (IRMAR)
	CNRS : UMR6625 – Université de Rennes I – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées de Rennes

Subject

:

Mathematics/Geometric Topology Mathematics/Group Theory

Keyword(s) : braid – curve diagram – complexity – lamination – Teichmüller space

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From: Bert Wiest <>

Submitted on: Friday, 14 April 2006 17:36:47

Updated on: Wednesday, 11 March 2009 09:28:57