Meyerhoff manifold

In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by ${\displaystyle (5,1)}$ surgery on the figure-8 knot complement. It was introduced by Robert Meyerhoff (1987) as a possible candidate for the hyperbolic 3-manifold of smallest volume, but the Weeks manifold turned out to have slightly smaller volume. It has the second smallest volume

${\displaystyle V_{m}=12\cdot (283)^{3/2}\zeta _{k}(2)(2\pi )^{-6}=0.981368\dots }$

of orientable arithmetic hyperbolic 3-manifolds, where ${\displaystyle \zeta _{k}}$ is the zeta function of the quartic field of discriminant ${\displaystyle -283}$. Alternatively,

${\displaystyle V_{m}=\Im ({\rm {{Li}_{2}(\theta )+\ln |\theta |\ln(1-\theta ))=0.981368\dots }}}$

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 quartic ${\displaystyle \theta ^{4}+\theta -1=0}$.

Ted Chinburg (1987) showed that this manifold is arithmetic.