Connection between Tomboulis vortices and projection vortices

By using the freedom of picking a representative we explore connections between the Tomboulis SO(3)xZ(2) form of the partition function and the SU(2) form. We are able to express the monopole and vortex observables of the former in terms of configurations of the latter. Also we can measure Tomboulis and projection vortex counters on the same configuration to search for correlations.