In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link. It has volume approximately equal to 0.942707… (OEIS: A126774) and David Gabai, Robert Meyerhoff, and Peter Milley (2009) showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by Jeffrey Weeks (1985) as well as Sergei V. Matveev and Anatoly T. Fomenko (1988).

Volume

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.

Related manifolds

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.