  The Frobenius-Schur indicator is a parameter $\kappa_a=\pm 1$ assigned to
each self-dual particle $a$ in a TQFT. If $\kappa_a$ is negative then
straightening out a timelike zig-zag in the worldline of a particle of type $a$
can incur a minus sign and in this case the amplitude associated with the
diagram is not invariant under deformation. This has caused some confusion
about the topological invariance of even simple theories to space-time
deformations. We clarify that, given a TQFT with negative Frobenius-Schur
indicators, there are two distinct conventions commonly used to interpret a
spacetime diagram as a physical amplitude, only one of which is isotopy
invariant. We clarify in what sense TQFTs based on Chern-Simons theory with
negative Frobenius-Schur indicators are isotopy invariant, and we explain how
the Frobenius-Schur indicator is intimately linked with the need to frame
world-lines in Chern-Simons theory. Further, in the non-isotopy-invariant
interpretation of the diagram algebra we show how a trick of bookkeeping can
usually be invoked to push minus signs onto the diagrammatic value of a loop
(the "loop weight"), such that most of the evaluation of a diagram does not
incur minus signs from straightening zig-zags, and only at the last step minus
signs are added. We explain the conditions required for this to be possible. We
then further examine what is required in order for a theory to have full
isotopy invariance of planar spacetime diagrams, and discover that, if we have
successfully pushed the signs from zig-zags onto the loop weight, the only
possible obstruction to this is given by an object related to vertices, known
as the "third Frobenius-Schur indicator". We finally discuss the extent to
which this gives us full isotopy invariance for braided theories.
