Vol. 221, No. 1, 2005

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Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups

John Crisp and Luis Paris

Vol. 221 (2005), No. 1, 1–27
Abstract

From a group H and h H, we define a representation ρ : Bn Aut(Hn), where Bn denotes the braid group on n strands, and Hn denotes the free product of n copies of H. We call ρ the Artin type representation associated to the pair (H,h). Here we study various aspects of such representations.

Firstly, we associate to each braid β a group Γ(H,h)(β) and prove that the operator Γ(H,h) determines a group invariant of oriented links. We then give a topological construction of the Artin type representations and of the link invariant Γ(H,h), and we prove that the Artin type representations are faithful if and only if h is nontrivial. The last part of the paper is devoted to the study of some semidirect products Hn ρBn, where ρ : Bn Aut(Hn) is an Artin type representation. In particular, we show that Hn ρBn is a Garside group if H is a Garside group and h is a Garside element of H.

Keywords
braid group, Garside group, link invariants
Mathematical Subject Classification 2000
Primary: 20F36
Secondary: 57M27, 20F10
Milestones
Received: 9 January 2003
Revised: 21 January 2004
Accepted: 21 January 2004
Published: 1 September 2005
Authors
John Crisp
Institut de Mathématiques de Bourgogne
Université de Bourgogne
UMR 5584 du CNRS, BP 47870
21078 Dijon Cedex
France
Luis Paris
Institut de Mathématiques de Bourgogne
Université de Bourgogne
UMR 5584 du CNRS, BP 47870
21078 Dijon Cedex
France