Local invariance under orientation and parity reversal

Establish whether the evaluation of type \(\mathsf{D}\) webs remains invariant under locally reversing the orientations and simultaneously swapping the parity of spin edges, in general.

Background

The paper proves a global invariance result: reversing all spin-edge orientations and, when appropriate, exchanging even and odd spin-edge parities leaves the web evaluation unchanged. The authors then consider the corresponding local operation, which would provide a stronger and more flexible invariance principle for type $\mathsf{D}$ web evaluations.

Although the local operation is verified in at least one example through the singular-square relation, the authors do not establish it for arbitrary webs. Determining whether this local invariance holds generally would clarify the extent to which the global symmetry can be localized and could reduce the number of cases required in proving skein relations.