Lawrence-Krammer-Bigelow representations and dual Garside length of braids

Tetsuya Ito (1) , Bert Wiest (2)

1 Graduate School of Mathematical Sciences[Tokyo]
2 IRMAR - Institut de Recherche Mathématique de Rennes

We show that the span of the variable $q$ in the Lawrence-Krammer-Bigelow representation matrix of a braid is equal to the twice of the dual Garside length of the braid, as was conjectured by Krammer. Our proof is close in spirit to Bigelow's geometric approach. The key observation is that the dual Garside length of a braid can be read off a certain labeling of its curve diagram.

Mots clés: Lawrence-Krammer-Bigelow representation; braid group; curve diagram; dual Garside length

Domaines: Théorie des groupes [math.GR]; Topologie géométrique [math.GT]

Tetsuya Ito, Bert Wiest. Lawrence-Krammer-Bigelow representations and dual Garside length of braids. Geometry and Topology, 2015, 19 (3), pp.1361-1381. ⟨10.2140/gt.2015.19.1361⟩. ⟨hal-00656480⟩