  Counterterms that are not reducible to $\zeta_{n}$ are generated by ${}_3F_2$
hypergeometric series arising from diagrams for which triangle and uniqueness
relations furnish insufficient data. Irreducible double sums, corresponding to
the torus knots $(4,3)=8_{19}$ and $(5,3)=10_{124}$, are found in anomalous
dimensions at ${\rm O}(1/N^3)$ in the large-$N$ limit, which we compute
analytically up to terms of level 11, corresponding to 11 loops for
4-dimensional field theories and 12 loops for 2-dimensional theories.
High-precision numerical results are obtained up to 24 loops and used in Pad\'e
resummations of $\varepsilon$-expansions, which are compared with analytical
results in 3 dimensions. The ${\rm O}(1/N^3)$ results entail knots generated by
three dressed propagators in the master two-loop two-point diagram. At higher
orders in $1/N$ one encounters the uniquely positive hyperbolic 11-crossing
knot, associated with an irreducible triple sum. At 12 crossings, a pair of
3-braid knots is generated, corresponding to a pair of irreducible double sums
with alternating signs. The hyperbolic positive knots $10_{139}$ and $10_{152}$
are not generated by such self-energy insertions.
