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Representations of the Necklace Braid Group: Topological and Combinatorial Approaches

Abstract

The necklace braid group \({\mathcal {NB}}_n\) is the motion group of the \(n+1\) component necklace link \(\mathcal {L}_n\) in Euclidean \(\mathbb {R}^3\). Here \(\mathcal {L}_n\) consists of n pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group \({\mathcal {NB}}_n\), especially those obtained as extensions of representations of the braid group \(\mathcal {B}_n\) and the loop braid group \({\mathcal {LB}}_n\). We show that any irreducible \(\mathcal {B}_n\) representation extends to \({\mathcal {NB}}_n\) in a standard way. We also find some non-standard extensions of several well-known \(\mathcal {B}_n\)-representations such as the Burau and LKB representations. Moreover, we prove that any local representation of \(\mathcal {B}_n\) (i.e., coming from a braided vector space) can be extended to \({\mathcal {NB}}_n\), in contrast to the situation with \({\mathcal {LB}}_n\). We also discuss some directions for future study from categorical and physical perspectives.

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Correspondence to Eric C. Rowell.

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ER and AK gratefully acknowledge support under USA NSF Grant DMS-1664359, and the support of Texas A&M through a Presidential Impact Fellowship. AB and PM thank EPSRC for support under Grant EP/I038683/1, and PM also acknowledges support from the Leverhulme Trust. PM and ER thank Celeste Damiani for useful conversations. Part of this paper was written during ER’s visit to BICMR, Peking University, and he gratefully acknowledges the support of that institution.

Communicated by Y. Kawahigashi

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Bullivant, A., Kimball, A., Martin, P. et al. Representations of the Necklace Braid Group: Topological and Combinatorial Approaches. Commun. Math. Phys. (2019) doi:10.1007/s00220-019-03445-0

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