In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by ( 5 , 1 ) {\displaystyle (5,1)} {\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 \( \zeta _{k} \) 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 |x| is the absolute value of the complex root\( \theta \) (with positive imaginary part) of the quartic \( {\displaystyle \theta ^{4}+\theta -1=0} \).

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

See also

Gieseking manifold

Weeks manifold

References

Chinburg, Ted (1987), "A small arithmetic hyperbolic three-manifold", Proceedings of the American Mathematical Society, 100 (1): 140–144, doi:10.2307/2046135, ISSN 0002-9939, JSTOR 2046135, MR 0883417

Chinburg, Ted; Friedman, Eduardo; Jones, Kerry N.; Reid, Alan W. (2001), "The arithmetic hyperbolic 3-manifold of smallest volume", Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, 30 (1): 1–40, ISSN 0391-173X, MR 1882023

Meyerhoff, Robert (1987), "A lower bound for the volume of hyperbolic 3-manifolds", Canadian Journal of Mathematics, 39 (5): 1038–1056, doi:10.4153/CJM-1987-053-6, ISSN 0008-414X, MR 0918586

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