Meyerhoff manifold
In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by [math]\displaystyle{ (5,1) }[/math] 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
- [math]\displaystyle{ V_m = 12\cdot(283)^{3/2}\zeta_k(2)(2\pi)^{-6} = 0.981368\dots }[/math]
of orientable arithmetic hyperbolic 3-manifolds, where [math]\displaystyle{ \zeta_k }[/math] is the zeta function of the quartic field of discriminant [math]\displaystyle{ -283 }[/math]. Alternatively,
- [math]\displaystyle{ V_m = \Im(\rm{Li}_2(\theta)+\ln|\theta|\ln(1-\theta)) = 0.981368\dots }[/math]
where [math]\displaystyle{ \rm{Li}_n }[/math] is the polylogarithm and [math]\displaystyle{ |x| }[/math] is the absolute value of the complex root [math]\displaystyle{ \theta }[/math] (with positive imaginary part) of the quartic [math]\displaystyle{ \theta^4+\theta-1=0 }[/math].
Ted Chinburg (1987) showed that this manifold is arithmetic.
See also
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
- 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, 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
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