Meyerhoff manifold

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Short description: Mathemical concept

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

Vm=12(283)3/2ζk(2)(2π)6=0.981368

of orientable arithmetic hyperbolic 3-manifolds, where ζk is the zeta function of the quartic field of discriminant 283. Alternatively,

Vm=(Li2(θ)+ln|θ|ln(1θ))=0.981368

where Lin is the polylogarithm and |x| is the absolute value of the complex root θ (with positive imaginary part) of the quartic θ4+θ1=0.

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

See also

References