The Burau representations of loop braid groups

Martin Palmer; Arthur Soulié

doi:10.5802/crmath.338

Géométrie et Topologie, Théorie des représentations

The Burau representations of loop braid groups

Martin Palmer ¹ ; Arthur Soulié ²

¹ Institutul de Matematică Simion Stoilow al Academiei Române, 21 Calea Griviței, 010702 București, România
² University of Glasgow, School of Mathematics and Statistics, 132 University Pl, Glasgow G12 8TA, United Kingdom

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 781-797.

Résumé
Abstract

Nous donnons une construction topologique simple et naturelle des représentations de Burau des groupes de tresses soudées. Il en existe en fait quatre versions : ces représentations peuvent être définies pour les groupes de tresses soudées étendues ou non étendues, et dans ces deux cas, il y a une version réduite et une autre non réduite. Pour trois d’entre elles, d’un point de vue rigoureusement algébrique, on peut aisément déterminer les matrices correspondant aux générateurs des groupes considérés. En revanche, la quatrième est plus subtile et ne semble pas évidente à déterminer d’un strict point de vue combinatoire, alors qu’elle est topologiquement très naturelle à définir.

We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced version. Three are not surprising, and one could easily guess the correct matrices to assign to generators. The fourth is more subtle, and does not seem combinatorially obvious, although it is topologically very natural.

Reçu le : 2021-10-04
Accepté le : 2022-01-26
Publié le : 2022-06-30

DOI : 10.5802/crmath.338

Classification : 20C12, 20F36, 20J05, 57M07, 57M10

Affiliations des auteurs :

Martin Palmer ¹ ; Arthur Soulié ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Martin Palmer and Arthur Souli\'e},
     title = {The {Burau} representations of loop braid groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {781--797},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.338},
     language = {en},
}

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Martin Palmer; Arthur Soulié. The Burau representations of loop braid groups. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 781-797. doi : 10.5802/crmath.338. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.338/

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