  Given a fully extended topological quantum field theory, the 'crossing with
the circle' conditions establish that the dimension, or categorification
thereof, of the quantum invariant assigned to a closed $k$-manifold $\Sigma$ is
equivalent to that assigned to the ($k$+1)-manifold $\Sigma \times \mathbb
S^1$. We compute in this manuscript these conditions for the 4-3-2-1
Dijkgraaf-Witten theory. In the context of the lattice Hamiltonian realisation
of the theory, the quantum invariants assigned to the circle and the torus
encode the defect open string-like and bulk loop-like excitations,
respectively. The corresponding 'crossing with the circle' condition thus
formalises the process by which loop-like excitations are formed out of
string-like ones. Exploiting this result, we revisit the statement that
loop-like excitations define representations of the linear necklace group as
well as the loop braid group.
