  We simplify, to a single integral of dilogarithms, the least tractable
O(1/N^3) contribution to the large-N critical exponent $\eta$ of the non-linear
sigma-model, and hence $\phi^4$-theory, for any spacetime dimensionality, D. It
is the sole generator of irreducible multiple zeta values in epsilon-expansions
with $D=2-2\epsilon$, for the sigma-model, and $D=4-2\epsilon$, for
$\phi^4$-theory. In both cases we confirm results of Broadhurst, Gracey and
Kreimer (BGK) that relate knots to counterterms. The new compact form is much
simpler than that of BGK. It enables us to develop 8 new terms in the
epsilon-expansion with $D=3-2\epsilon$. These involve alternating Euler sums,
for which the basis of irreducibles is larger. We conclude that massless
Feynman diagrams in odd spacetime dimensions share the greater transcendental
complexity of massive diagrams in even dimensions, such as those contributing
to the electron's magnetic moment and the electroweak $\rho$-parameter.
Consequences for the perturbative sector of Chern-Simons theory are discussed.
