Weeks manifold

Short description: Smallest closed orientable hyperbolic 3-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,

$V_w=\mathrm{\Im }({\mathrm{L}\mathrm{i}}_{\mathrm{2}}\mathrm{(}\mathrm{\theta }\mathrm{)}\mathrm{+}\ln \mathrm{|}\mathrm{\theta }\mathrm{|}\ln \mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\theta }\mathrm{)}\mathrm{)}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{9}\mathrm{4}\mathrm{2}\mathrm{7}\mathrm{0}\mathrm{7}\mathrm{\dots }$

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 cubic.

Symmetries

The Weeks manifold has symmetry group $D_6$, the dihedral group of order 12. Quotients by this group and its subgroups can be used to characterize the manifold as a branched covering based on an orbifold. In particular, the quotient by the order-3 subgroup of the symmetry group has underlying set a 3-sphere and branch set a 5₂ knot.

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 – another manifold with very 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