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Given a fully extended topological quantum field theory, the 'crossing withthe circle' conditions establish that the dimension, or categorificationthereof, of the quantum invariant assigned to a closed $k$-manifold $\Sigma$ isequivalent to that assigned to the ($k$+1)-manifold $\Sigma \times \mathbbS^1$. We compute in this manuscript these conditions for the 4-3-2-1Dijkgraaf-Witten theory. In the context of the lattice Hamiltonian realisationof the theory, the quantum invariants assigned to the circle and the torusencode the defect open string-like and bulk loop-like excitations,respectively. The corresponding 'crossing with the circle' condition thusformalises the process by which loop-like excitations are formed out ofstring-like ones. Exploiting this result, we revisit the statement thatloop-like excitations define representations of the linear necklace group aswell as the loop braid group.

Crossing with the circle in Dijkgraaf-Witten theory and applications to topological phases of matter