# Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links

Borwein, Jonathan M. and Broadhurst, David J. (1998) Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links. [Preprint]

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We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $$\frac{a}{b}\,{\rm vol}({\cal M})=\frac{(-D)^{3/2}}{(2\pi)^{2n-4}}\,\frac {\zeta_K(2)}{2\zeta(2)}$$ for a manifold ${\cal M}$ 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.