Meyerhoff manifold

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 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

$12\cdot(283)^{3/2}\zeta_k(2)(2\pi)^{-6} = 0.9812\ldots$

of orientable arithmetic hyperbolic 3-manifolds (where ζ_k is the zeta function of the quartic field of discriminant −283). Chinburg (1987) showed that it is arithmetic.

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, MR883417 
• 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, MR1882023, http://www.numdam.org/item?id=ASNSP_2001_4_30_1_1_0 
• 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, MR918586