# Every braid admits a short sigma-definite expression

Journal of the European Mathematical Society (2011)

- Volume: 013, Issue: 6, page 1591-1631
- ISSN: 1435-9855

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topFromentin, Jean. "Every braid admits a short sigma-definite expression." Journal of the European Mathematical Society 013.6 (2011): 1591-1631. <http://eudml.org/doc/277298>.

@article{Fromentin2011,

abstract = {A result by Dehornoy (1992) says that every nontrivial braid admits a $\sigma $-definite expression, defined as a braid word in which the generator $\sigma _i$ with maximal index $i$ appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we
enhance this result and prove that every braid admits a $\sigma $-definite word expression that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new normal form called the rotating normal form.},

author = {Fromentin, Jean},

journal = {Journal of the European Mathematical Society},

keywords = {braid group; braid ordering; dual braid monoid; normal form; braid groups; braid ordering; dual braid monoid; normal form theorems; algorithms; sigma-definite expression; Artin generators; Birman-Ko-Lee generators},

language = {eng},

number = {6},

pages = {1591-1631},

publisher = {European Mathematical Society Publishing House},

title = {Every braid admits a short sigma-definite expression},

url = {http://eudml.org/doc/277298},

volume = {013},

year = {2011},

}

TY - JOUR

AU - Fromentin, Jean

TI - Every braid admits a short sigma-definite expression

JO - Journal of the European Mathematical Society

PY - 2011

PB - European Mathematical Society Publishing House

VL - 013

IS - 6

SP - 1591

EP - 1631

AB - A result by Dehornoy (1992) says that every nontrivial braid admits a $\sigma $-definite expression, defined as a braid word in which the generator $\sigma _i$ with maximal index $i$ appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we
enhance this result and prove that every braid admits a $\sigma $-definite word expression that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new normal form called the rotating normal form.

LA - eng

KW - braid group; braid ordering; dual braid monoid; normal form; braid groups; braid ordering; dual braid monoid; normal form theorems; algorithms; sigma-definite expression; Artin generators; Birman-Ko-Lee generators

UR - http://eudml.org/doc/277298

ER -

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