Weeks manifold
In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link. It has volume approximately equal to 0.942707… (OEIS: A126774) and David Gabai, Robert Meyerhoff, and Peter Milley (2009) showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by Jeffrey Weeks (1985) as well as Sergei V. Matveev and Anatoly T. Fomenko (1988).
Volume
Since the Weeks manifold is an arithmetic hyperbolic 3-manifold, its volume can be computed using its arithmetic data and a formula due to Armand Borel:
- [math]\displaystyle{ V_w = \frac{3 \cdot23^{3/2}\zeta_k(2)}{4\pi^4} = 0.942707\dots }[/math]
where [math]\displaystyle{ k }[/math] is the number field generated by [math]\displaystyle{ \theta }[/math] satisfying [math]\displaystyle{ \theta^3-\theta+1=0 }[/math] and [math]\displaystyle{ \zeta_k }[/math] is the Dedekind zeta function of [math]\displaystyle{ k }[/math]. [1] Alternatively,
- [math]\displaystyle{ V_w = \Im(\rm{Li}_2(\theta)+\ln|\theta|\ln(1-\theta)) = 0.942707\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 cubic.
Related manifolds
The cusped hyperbolic 3-manifold obtained by (5, 1) Dehn surgery on the Whitehead link is the so-called sibling manifold, or sister, of the figure-eight knot complement. The figure eight knot's complement and its sibling have the smallest volume of any orientable, cusped hyperbolic 3-manifold. Thus the Weeks manifold can be obtained by hyperbolic Dehn surgery on one of the two smallest orientable cusped hyperbolic 3-manifolds.
See also
- Meyerhoff manifold - second small volume
References
- "Lower bounds on volumes of hyperbolic Haken 3-manifolds (with an appendix by Nathan Dunfield)", Journal of the American Mathematical Society 20 (4): 1053–1077, 2007, doi:10.1090/S0894-0347-07-00564-4, Bibcode: 2007JAMS...20.1053A.
- 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, http://www.numdam.org/item?id=ASNSP_2001_4_30_1_1_0
- Gabai, David; Meyerhoff, Robert; Milley, Peter (2009), "Minimum volume cusped hyperbolic three-manifolds", Journal of the American Mathematical Society 22 (4): 1157–1215, doi:10.1090/S0894-0347-09-00639-0, Bibcode: 2009JAMS...22.1157G
- Matveev, Sergei V.; Fomenko, Aanatoly T. (1988), "Isoenergetic surfaces of Hamiltonian systems, the enumeration of three-dimensional manifolds in order of growth of their complexity, and the calculation of the volumes of closed hyperbolic manifolds", Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 43 (1): 5–22, doi:10.1070/RM1988v043n01ABEH001554, Bibcode: 1988RuMaS..43....3M
- Weeks, Jeffrey (1985), Hyperbolic structures on 3-manifolds, Ph.D. thesis, Princeton University
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