Geometry & Topology

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

Tetsuya Ito and Bertold Wiest

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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.

Article information

Geom. Topol., Volume 19, Number 3 (2015), 1361-1381.

Received: 13 December 2013
Accepted: 26 August 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F36: Braid groups; Artin groups
Secondary: 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] 57M07: Topological methods in group theory

Lawrence-Krammer-Bigelow representation braid group curve diagram dual Garside length


Ito, Tetsuya; Wiest, Bertold. Lawrence–Krammer–Bigelow representations and dual Garside length of braids. Geom. Topol. 19 (2015), no. 3, 1361--1381. doi:10.2140/gt.2015.19.1361.

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