Volume 19, issue 3 (2015)

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

Tetsuya Ito and Bert Wiest

Geometry & Topology 19 (2015) 1361–1381
Abstract

We show that the span of the variable $q$ in the Lawrence–Krammer–Bigelow representation matrix of a braid is equal to twice 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.

Keywords
Lawrence-Krammer-Bigelow representation, braid group, curve diagram, dual Garside length
Mathematical Subject Classification 2010
Primary: 20F36
Secondary: 20F10, 57M07