Crossing symmetry including non planar diagrams in perturbative QFT

We venture a proof of crossing symmetry for non-planar diagrams in perturbative QFT. For the planar diagrams a proof of crossing is available in the literature and our method closely follows the one depicted in that case. We classify the non-planar diagrams broadly into two types. For one of these types the proof is pretty straightforward and hence the result extends to all point all loop on-shell amplitudes. These are called the "trivial" cases while for the other type we find certain cases called the "non trivial" cases for which the proof is much more subtle. We present an explicit example of such a "non trivial" case at 3-loop order and argue how the proof of crossing symmetry holds true when all subtleties are taken into consideration. Although a general proof for these "nontrivial cases" at higher loop is beyond the scope of this paper, we do conjecture that the proof of crossing shall hold in a similar fashion, for higher loops as well since the subtlety of the resolution in 3-loop case lies with the type of Landau singularity we need to consider and not the number of internal propagators or loop momenta.