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We studied in [Ann. Inst. Henri Poincar\'e D 9, 367-433, (2022)], a complexmulti-matrix model with $\mathrm{U}(N)^2 \times \mathrm{O}(D)$ symmetry, andwhose double scaling limit where simultaneously the large-$N$ and large-$D$limits were taken while keeping the ratio $N/\sqrt{D}=M$ finite and fixed. Inthis double scaling limit, the complete recursive characterization of theFeynman graphs of arbitrary genus for the leading order grade $\ell=0$ wasachieved. In this current study, we classify the higher order graphs in $\ell$.More specifically, $\ell=1$ and $\ell=2$ with arbitrary genus, in addition to aspecific class of two-particle-irreducible (2PI) graphs for higher $\ell\geqslant 3$ but with genus zero. Furthermore, we demonstrate that counting the2PI graphs with a single $\mathrm{O}(D)$-loop corresponds to enumerating thealternating knots using the Rolsen's table results, performing a connected sum,or Tait flyping moves on them.

Classification of higher grade $\ell$ graphs for $\mathrm{U}(N)^2\times \mathrm{O}(D)$ multi-matrix models