  We evaluate all the primitive divergences contributing to the 7--loop
$\beta$\/--function of $\phi^4$ 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 $10_{124}$,
$10_{139}$, and $10_{152}$, 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 $10^{11}$. The series of `zig--zag' counterterms,
$\{6\zeta_3,\,20\zeta_5,\, \frac{441}{8}\zeta_7,\,168\zeta_9,\,\ldots\}$,
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 term is $4C_{n-1}
\sum_{p>0}\frac{(-1)^{p n - n}}{p^{2n-3}}$, where $C_n=\frac{1}{n+1}{2n \choose
n}$ are the Catalan numbers, familiar in knot theory. The investigations
reported here entailed intensive use of REDUCE, to generate ${\rm O}(10^4)$
lines of code for multiple precision FORTRAN computations, enabled by Bailey's
MPFUN routines, running for ${\rm O}(10^3)$ CPUhours on DecAlpha machines.
