Non-planar spin bits beyond two loops
  We study higher-loop orders of spin bit models underlying the non-planar dynamics of $\mathcal{N}=4$ SYM gauge theory. In particular, we derive a ''tower'' of non-planar identities involving products of site permutation operators. Such identities are then applied in the formulation of planarly consistent, testable conjectures for the full non-planar, higher-loop Hamiltonian of the $su(2)$ spin-chain. 
S. Bellucci, A. Marrani