A note on representations of welded braid groups

Abstract

In this paper, we adapt the procedure of the Long-Moody procedure to construct linear representations of welded braid groups. We exhibit the natural setting in this context and compute the first examples of representations we obtain thanks to this method. We take this way also the opportunity to review the few known linear representations of welded braid groups.

Unrestricted virtual braids and crystallographic braid groups

“…where UC stands for 'Under Commuting', do not hold in W B n (see for instance [7,21]). Consequently, the following group, which was first defined by Kauffman and Lambropoulou in [22] As in the case of V B n , let π U V P : U V B n −→ S n be the homomorphism defined by π U V P (σ i ) = π U V P (ρ i ) = s i for i = 1, .…”

The Long-Moody construction and twisted Alexander invariants

“…However we only use the original construction of Long in this paper, and thus we write it simply without these homomorphisms. Also, Bellingeri and Soulié [1] defined this contruction for the welded braid group W B n which is a generalization of the braid group.…”

Finite image homomorphisms of the braid group and its generalizations

“…The welded braid group. The welded braid group, wB n , is a quotient of vB n by the Over Crossings Commute relation, or "OC" relation, defined as τ i σ i+1 τ i = σ i+1 σ i τ i+1 [3].…”