Beyond the triangle and uniqueness relations: non-zeta counterterms at large N from positive knots

Counterterms that are not reducible to zeta(n) are generated by F-3(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 O(1/N-3) in the large-N limit, which we compute analytically up to terms of level II, corresponding to Il 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 Pade resummations of E-expansions, which are compared with analytical results in 3 dimensions. The 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 Ii-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.