A topological correctness criterion for non-commutative logic

We formulate Girard's long trip criterion for multiplicative linear logic (MLL) in a topological way, by associating a ribbon diagram to every switching, and requiring that it is homeomorphic to the disk. Then, we extend the well-known planarity criterion for multiplicative cyclic linear logic (McyLL) to multiplicative non-commutative logic (MNL) and show that the resulting planarity criterion is equivalent to Abrusci and Ruet's original long trip criterion for MNL