KNOTS AND NUMBERS IN PHI(4) THEORY TO 7 LOOPS AND BEYOND

We evaluate all the primitive divergences contributing to the 7-loop b eta-function of phi(4) theory, i.e. all 59 diagrams that are free of s ubdivergences and hence give scheme-independent contributions. Guided by the association of diagrams with knots, we obtain analytical result s for 56 diagrams. The remaining three diagrams, associated with the k nots 10(124), 10(139), and 10(152), are evaluated numerically, to 10 s f. Only one satellite knot with 11 crossings is encountered and the tr anscendental number associated with it is found. Thus we achieve an an alytical result for the 6-loop contributions, and a numerical result a t 7 loops that is accurate to one part in 10(11). The series of 'zig-z ag' counterterms, {6 zeta(3), 20 zeta(5), 441/8 zeta 7, 168 zeta 9,... }, previously known for n = 3, 4, 5, 6 loops, is evaluated to 10 loops , corresponding to 17 crossings, revealing that the n-loop zigzag term is 4C(n-1) Sigma(p>0) (-1)(pn-n)/p(2n-3), where C-n = 1/n+1((2n)(n)) are the Catalan numbers, familiar in knot theory. The investigations r eported here entailed intensive use of REDUCE, to generate O(10(4)) li nes of code for multiple precision FORTRAN computations, enabled by Ba iley's MPFUN routines, running for O(10(3)) CPUhours on DecAlpha machi nes.