  3+1 dimensional topological phases can support loop-like excitations in
addition to point-like ones, allowing for non-trivial loop-loop and point-loop
braiding statistics not permitted to point-like excitations alone. Furthermore,
these loop-like excitations can be linked together, changing their properties.
In particular, this can lead to distinct three-loop braiding, involving two
loops undergoing an exchange process while linked to a third loop. In this
work, we investigate the loop-like excitations in a 3+1d Hamiltonian
realization of Dijkgraaf-Witten theory through direct construction of their
membrane operators, for a general finite Abelian group and 4-cocycle twist.
Using these membrane operators, we find the braiding relations and fusion rules
for the loop-like excitations, including those linked to another loop-like
excitation. Furthermore, we use these membrane operators to construct
projection operators that measure the topological charge and show that the
number of distinct topological charges measured by the 2-torus matches the
ground state degeneracy of the model on the 3-torus, explicitly confirming a
general expectation for topological phases. This direct construction of the
membrane operators sheds significant light on the key properties of the
loop-like excitations in 3+1 dimensional topological phases.
