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In this work we have studied the Kleiss-Kuijf relations for the recentlyintroduced Parke-Taylor factors at one-loop in the CHY approach, that reproducequadratic Feynman propagators. By doing this, we were able to identify thenon-planar one-loop Parke-Taylor factors. In order to check that, in fact,these new factors can describe non-planar amplitudes, we applied them to thebi-adjoint $\Phi^3$ theory. As a byproduct, we found a new type of graphs thatwe called the non-planar CHY-graphs. These graphs encode all the informationfor the subleading order at one-loop, and there is not an equivalent of thesein the Feynman formalism.

Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators