Meyerhoff manifold

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

$V_m=12\cdot (283{)}^{3/2}{\zeta }_k(2)(2\pi {)}^{-6}=0.981368\dots$

of orientable arithmetic hyperbolic 3-manifolds, where ${\zeta }_k$ is the zeta function of the quartic field of discriminant $-283$. Alternatively,

$V_m=\mathrm{\Im }({\mathrm{L}\mathrm{i}}_{\mathrm{2}}\mathrm{(}\mathrm{\theta }\mathrm{)}\mathrm{+}\ln \mathrm{|}\mathrm{\theta }\mathrm{|}\ln \mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\theta }\mathrm{)}\mathrm{)}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{9}\mathrm{8}\mathrm{1}\mathrm{3}\mathrm{6}\mathrm{8}\mathrm{\dots }$

where ${\mathrm{L}\mathrm{i}}_{\mathrm{n}}$ is the polylogarithm and $|x|$ is the absolute value of the complex root $\theta$ (with positive imaginary part) of the quartic ${\theta }^4+\theta -1=0$.

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 
• 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