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.

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