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 un iqueness relations furnish insufficient data. irreducible double sums, corresponding to the torus knots (4,3) = 8(19) and (5,3) = 10(124), a re found in anomalous dimensions at O(1/N-3) in the large-N limit, whi ch we compute analytically up to terms of level II, corresponding to I l loops for 4-dimensional field theories and 12 loops for 2-dimensiona l theories. High-precision numerical results are obtained up to 24 loo ps and used in Pade resummations of E-expansions, which are compared w ith analytical results in 3 dimensions. The O(1/N-3) results entail kn ots generated by three dressed propagators in the master two-loop two- point diagram. At higher orders in 1/N one encounters the uniquely pos itive hyperbolic Ii-crossing knot, associated with an irreducible trip le sum. At 12 crossings, a pair of 3-braid knots is generated, corresp onding to a pair of irreducible double sums with alternating signs. Th e hyperbolic positive knots 10(139) and 10(152) are not generated by s uch self-energy insertions.