Knots and Numbers in φ 4 Theory to 7 Loops and Beyond

We evaluate all the primitive divergences contributing to the 7–loop β–function of φ4 theory, i.e. all 59 diagrams that are free of subdivergences and hence give scheme– independent contributions. Guided by the association of diagrams with knots, we obtain analytical results for 56 diagrams. The remaining three diagrams, associated with the knots 10124, 10139, and 10152, are evaluated numerically, to 10 sf. Only one satellite knot with 11 crossings is encountered and the transcendental number associated with it is found. Thus we achieve an analytical result for the 6–loop contributions, and a numerical result at 7 loops that is accurate to one part in 1011. The series of ‘zig–zag’ counterterms, {6ζ3, 20ζ5, 441 8 ζ7, 168ζ9, . . .}, previously known for n = 3, 4, 5, 6 loops, is evaluated to 10 loops, corresponding to 17 crossings, revealing that the n–loop zig–zag