  We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally
related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $a/b
vol(M)=(-D)^{3/2}/(2\pi)^{2n-4} (\zeta_K(2))/(2\zeta(2))$ for a manifold M
whose invariant trace field $K$ has a single complex place, discriminant $D$,
degree $n$, and Dedekind zeta value $\zeta_K(2)$. The largest numerator of the
998 invariants of Hodgson-Weeks manifolds is, astoundingly,
$a=2^4\times23\times37\times691 =9,408,656$; the largest denominator is merely
b=9. We also study the rational invariant a/b for single-complex-place cusped
manifolds, complementary to knots and links, both within and beyond the
Hildebrand-Weeks census. Within the censi, we identify 152 distinct Dedekind
zetas rationally related to volumes. Moreover, 91 census manifolds have volumes
reducible to pairs of these zeta values. Motivated by studies of Feynman
diagrams, we find a 10-component 24-crossing link in the case n=2 and D=-20. It
is one of 5 alternating platonic links, the other 4 being quartic. For 8 of 10
quadratic fields distinguished by rational relations between Dedekind zeta
values and volumes of Feynman orthoschemes, we find corresponding links.
Feynman links with D=-39 and D=-84 are missing; we expect them to be as
beautiful as the 8 drawn here. Dedekind-zeta invariants are obtained for knots
from Feynman diagrams with up to 11 loops. We identify a sextic 18-crossing
positive Feynman knot whose rational invariant, a/b=26, is 390 times that of
the cubic 16-crossing non-alternating knot with maximal D_9 symmetry. Our
results are secure, numerically, yet appear very hard to prove by analysis.

High Energy Physics - Theory;Classical Analysis and ODEs;Geometric Topology;Number Theory