Weeks manifold

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).

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 Weeks manifold has symmetry group ${\displaystyle D_{6}}$, the dihedral group of order 12. Quotients by this group and its subgroups can be used to characterize the manifold as a branched covering based on an orbifold. In particular, the quotient by the order-3 subgroup of the symmetry group has underlying set a 3-sphere and branch set a 5₂ knot. ²

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.