Weeks manifold

Since the Weeks manifold is an arithmetic hyperbolic 3-manifold, its volume can be computed using its arithmetic data and a formula due to Armand Borel:

${\displaystyle V_{w}={\frac {3\cdot 23^{3/2}\zeta _{k}(2)}{4\pi ^{4}}}=0.942707\dots }$

where ${\displaystyle k}$ is the number field generated by ${\displaystyle \theta }$ satisfying ${\displaystyle \theta ^{3}-\theta +1=0}$ and ${\displaystyle \zeta _{k}}$ is the Dedekind zeta function of ${\displaystyle k}$. [1] Alternatively,

${\displaystyle V_{w}=\Im ({\rm {{Li}_{2}(\theta )+\ln |\theta |\ln(1-\theta ))=0.942707\dots }}}$

where ${\displaystyle {\rm {{Li}_{n}}}}$ is the polylogarithm and ${\displaystyle |x|}$ is the absolute value of the complex root ${\displaystyle \theta }$ (with positive imaginary part) of the cubic.

The cusped hyperbolic 3-manifold obtained by (5, 1) Dehn surgery on the Whitehead link is the so-called sibling manifold, or sister, of the figure-eight knot complement. The figure eight knot's complement and its sibling have the smallest volume of any orientable, cusped hyperbolic 3-manifold. Thus the Weeks manifold can be obtained by hyperbolic Dehn surgery on one of the two smallest orientable cusped hyperbolic 3-manifolds.

• Meyerhoff manifold - second small volume

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• Weeks, Jeffrey (1985), Hyperbolic structures on 3-manifolds, Ph.D. thesis, Princeton University