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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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
References
Publication
Received: 13 December 2013
Accepted: 26 August 2014
Published: 21 May 2015
Proposed: Cameron Gordon
Seconded: Colin Rourke, Vaughan Jones
Authors
Tetsuya Ito
Research Institute for Mathematical Sciences
Kyoto University
Sakyo-ku
Kyoto, 606-8502
Japan
http://www.kurims.kyoto-u.ac.jp/~tetitoh/
Bert Wiest
IRMAR, UMR 6625 du CNRS
Université de Rennes 1
Campus de Beaulieu
35042 Rennes Cedex
France
http://perso.univ-rennes1.fr/bertold.wiest