  Motivated by physical and topological applications, we study representations
of the group $\mathcal{LB}_3$ of motions of $3$ unlinked oriented circles in
$\mathbb{R}^3$. Our point of view is to regard the three strand braid group
$\mathcal{B}_3$ as a subgroup of $\mathcal{LB}_3$ and study the problem of
extending $\mathcal{B}_3$ representations. We introduce the notion of a
\emph{standard extension} and characterize $\mathcal{B}_3$ representations
admiting such an extension. In particular we show, using a classification
result of Tuba and Wenzl, that every irreducible $\mathcal{B}_3$ representation
of dimension at most $5$ has a (standard) extension. We show that this result
is sharp by exhibiting an irreducible $6$-dimensional $\mathcal{B}_3$
representation that has no extensions (standard or otherwise). We obtain
complete classifications of (1) irreducible $2$-dimensional $\mathcal{LB}_3$
representations (2) extensions of irreducible $3$-dimensional $\mathcal{B}_3$
representations and (3) irreducible $\mathcal{LB}_3$ representations whose
restriction to $\mathcal{B}_3$ has abelian image.
