Meyerhoff manifold information

In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by $(5,1)$ 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

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

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

V_{m}=\Im ({\rm {{Li}_{2}(\theta )+\ln |\theta |\ln(1-\theta ))=0.981368\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 quartic $\theta ^{4}+\theta -1=0$ .

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

Meyerhoff manifold information

and 10 Related for: Meyerhoff manifold information

Meyerhoff manifold

Meyerhoff

List of manifolds

Weeks manifold

Hyperbolic volume

David Gabai

Hyperbolic Dehn surgery

SnapPea

2023 deaths in the United States

Geometry Festival