# On the complexity of braids

• Volume: 009, Issue: 4, page 801-840
• ISSN: 1435-9855

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## Abstract

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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 ${}_{ij}$, which are Garside-like half-twists involving strings $i$ through $j$, and by counting powered generators ${\Delta }_{ij}^{k}$ as $log\left(|k|+1\right)$ 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 Teichmüller 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.

## How to cite

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Dynnikov, Ivan, and Wiest, Bert. "On the complexity of braids." Journal of the European Mathematical Society 009.4 (2007): 801-840. <http://eudml.org/doc/277393>.

@article{Dynnikov2007,
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 $_\{ij\}$, which are Garside-like half-twists involving strings $i$ through $j$, and by counting powered generators $\Delta ^k_\{ij\}$ 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 Teichmüller 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.},
author = {Dynnikov, Ivan, Wiest, Bert},
journal = {Journal of the European Mathematical Society},
keywords = {braid; curve diagram; complexity; lamination; train track; Artin braid groups; generators and relations; mapping class groups; Artin generators; lengths of braids; curve diagrams; complexities; laminations; train tracks},
language = {eng},
number = {4},
pages = {801-840},
publisher = {European Mathematical Society Publishing House},
title = {On the complexity of braids},
url = {http://eudml.org/doc/277393},
volume = {009},
year = {2007},
}

TY - JOUR
AU - Dynnikov, Ivan
AU - Wiest, Bert
TI - On the complexity of braids
JO - Journal of the European Mathematical Society
PY - 2007
PB - European Mathematical Society Publishing House
VL - 009
IS - 4
SP - 801
EP - 840
AB - 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 $_{ij}$, which are Garside-like half-twists involving strings $i$ through $j$, and by counting powered generators $\Delta ^k_{ij}$ 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 Teichmüller 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.
LA - eng
KW - braid; curve diagram; complexity; lamination; train track; Artin braid groups; generators and relations; mapping class groups; Artin generators; lengths of braids; curve diagrams; complexities; laminations; train tracks
UR - http://eudml.org/doc/277393
ER -

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