Abstract
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 Σ is equivalent to that assigned to the (k+1)-manifold Σ × S¹. 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.