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:

$V_{w}={\frac {3\cdot 23^{3/2}\zeta _{k}(2)}{4\pi ^{4}}}=0.942707\dots$

where $k$ is the number field generated by $\theta$ satisfying $\theta ^{3}-\theta +1=0$ and $\zeta _{k}$ is the Dedekind zeta function of $k$. Alternatively,

$V_{w}=\Im ({\rm {{Li}_{2}(\theta )+\ln |\theta |\ln(1-\theta ))=0.942707\dots }}$

where ${\rm {{Li}_{n}}}$ is the polylogarithm and $|x|$ is the absolute value of the complex root $\theta$ (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

• Agol, Ian; Storm, Peter A.; Thurston, William P. (2007), "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, arXiv:math.DG/0506338, Bibcode:2007JAMS...20.1053A, doi:10.1090/S0894-0347-07-00564-4, MR 2328715.
• 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, MR 1882023
• Gabai, David; Meyerhoff, Robert; Milley, Peter (2009), "Minimum volume cusped hyperbolic three-manifolds", Journal of the American Mathematical Society, 22 (4): 1157–1215, arXiv:0705.4325, Bibcode:2009JAMS...22.1157G, doi:10.1090/S0894-0347-09-00639-0, MR 2525782
• 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, Bibcode:1988RuMaS..43....3M, doi:10.1070/RM1988v043n01ABEH001554, MR 0937017
• Weeks, Jeffrey (1985), Hyperbolic structures on 3-manifolds, Ph.D. thesis, Princeton University