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… and  showed that it has the smallest  volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by as well as.

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:


 * $$V_w = \frac{3 \cdot23^{3/2}\zeta_k(2)}{4\pi^4} = 0.942707\dots$$

where $$k$$ is the number field generated by $$\theta$$ satisfying $$\theta^3-\theta+1=0$$ and $$\zeta_k$$ is the Dedekind zeta function of $$k$$. Alternatively,


 * $$V_w = \Im(\rm{Li}_2(\theta)+\ln|\theta|\ln(1-\theta)) = 0.942707\dots$$

where $$\rm{Li}_n$$ is the polylogarithm and $$|x|$$ is the absolute value of the complex root $$\theta$$ (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.