Representations of the Necklace Braid Group

Abstract

The necklace braid group NBvn is the motion group of the n + 1 component necklace link Lvn in Euclidean R^3 . The link Lvn 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 NBvn, especially those obtained as extensions of representations of the braid group Bvn. During this study, we show that any completely reducible Bn representation extends to NBvn in a standard way. We also investigate non-standard extensions of several well-known Bvn-representations such as the Burau and Lawrence-Krammer-Bigelow representations. Moreover, we prove that any local representation of Bvn (i.e. coming from a braided vector space) can be extended to NBvn. Motivated by the extensions of these local representations, we investigate local representations of Bvn from the twisted tensor products of group algebras. We start by discussing the case of using the group Zv3 × Zv3, and even give some explicit examples.