  We 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 counting the
2PI graphs with a single $\mathrm{O}(D)$-loop corresponds to enumerating the
alternating knots using the Rolsen's table results, performing a connected sum,
or Tait flyping moves on them.
