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

The authors studied in [Ann. Inst. Henri Poincar\'e D 9, 367-433, (2022)], a complex multi-matrix model with $\mathrm{U}(N)^2 \times \mathrm{O}(D)$ symmetry, and whose 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. In this double scaling limit, the complete recursive characterization of the Feynman graphs of arbitrary genus for the leading order grade $\ell=0$ was achieved. 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 a specific class of two-particle-irreducible (2PI) graphs for higher $\ell \geqslant 3$ but with genus zero. Furthermore, we demonstrate that each 2PI graph with a single $\mathrm{O}(D)$-loop with an arbitrary $\ell$ corresponds to a reduced alternating knot diagram with $\ell$ crossings as listed in the Rolfsen knot table, or a resulting alternating knot diagram obtained after performing the Tait flyping moves. We generalize to 2PR by considering the connected sum and the Reidemeister move I.