Geometry & Topology

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

Tetsuya Ito and Bertold Wiest

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation

References

  • S J Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001) 471–486
  • S J Bigelow, The Lawrence–Krammer representation, from: “Topology and geometry of manifolds”, (G Matić, C McCrory, editors), Proc. Sympos. Pure Math. 71, Amer. Math. Soc. (2003) 51–68
  • S J Bigelow, Homological representations of the Iwahori–Hecke algebra, from: “Proceedings of the Casson Fest”, (C Gordon, Y Rieck, editors), Geom. Topol. Monogr. 7 (2004) 493–507
  • J Birman, K H Ko, S J Lee, A new approach to the word and conjugacy problems in the braid groups, Adv. Math. 139 (1998) 322–353
  • T Ito, B Wiest, Erratum to “How to read the length of a braid from its curve diagram” [MR2813531], Groups Geom. Dyn. 7 (2013) 495–496
  • D Krammer, The braid group $B\sb 4$ is linear, Invent. Math. 142 (2000) 451–486
  • D Krammer, Braid groups are linear, Ann. of Math. 155 (2002) 131–156
  • R J Lawrence, Homological representations of the Hecke algebra, Comm. Math. Phys. 135 (1990) 141–191
  • L Paoluzzi, L Paris, A note on the Lawrence–Krammer–Bigelow representation, Algebr. Geom. Topol. 2 (2002) 499–518
  • B Wiest, How to read the length of a braid from its curve diagram, Groups Geom. Dyn. 5 (2011) 673–681