Geometry & Topology

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

Tetsuya Ito and Bertold Wiest

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510858764

Digital Object Identifier
doi:10.2140/gt.2015.19.1361

Mathematical Reviews number (MathSciNet)
MR3352238

Zentralblatt MATH identifier
1345.20049

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.gt/1510858764


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References

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