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… ({{OEIS2C|A126774}}) and {{harvs|txt|last1=Gabai|first1=David|authorlink1=David Gabai|last2=Meyerhoff|first2=Robert | last3=Milley | first3=Peter |year=2009}} showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by {{harvs|txt|last=Weeks|first=Jeffrey |authorlink=Jeffrey Weeks (mathematician)|year=1985}} as well as {{harvs|txt|last1=Matveev | first1=Sergei V. | last2=Fomenko | first2=Anatoly T. | author2-link=Anatoly Fomenko |year=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:
:
where is the number field generated by satisfying and is the Dedekind zeta function of . {{harvs | last1=Chinburg | first1=Ted | last2=Friedman | first2=Eduardo | last3=Jones | first3=Kerry N. | last4=Reid | first4=Alan W. | title=The arithmetic hyperbolic 3-manifold of smallest volume | mr=1882023 |zbl = 1008.11015 | year=2001 | journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV | volume=30 | issue=1 | pages=1–40}} Alternatively,
:
where is the polylogarithm and is the absolute value of the complex root (with positive imaginary part) of the cubic.
Symmetries
The Weeks manifold has symmetry group , 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. {{cite journal
|last1=Mednykh
|first1=Alexander
|last2=Vesnin
|first2=Andrei
|date=1998
|title=Visualization of the isometry group action on the Fomenko–Matveev–Weeks manifold
|journal=Journal of Lie Theory
|volume=8
|issue=1
|pages=51-66
|publisher=Heldermann Verlag
|doi=
|url=https://www.heldermann-verlag.de/jlt/jlt08/MEDVESLAT.PDF
|access-date=2025-05-28
}}
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
{{reflist}}
- {{citation
| last1 = Agol | first1 = Ian | author1-link=Ian Agol
| last2 = Storm | first2 = Peter A.
| last3 = Thurston | first3 = William P. | author3-link = William Thurston
| arxiv = math.DG/0506338
| doi = 10.1090/S0894-0347-07-00564-4
| mr = 2328715
| issue = 4
| journal = Journal of the American Mathematical Society
| pages = 1053–1077
| title = Lower bounds on volumes of hyperbolic Haken 3-manifolds (with an appendix by Nathan Dunfield)
| volume = 20
| year = 2007| bibcode = 2007JAMS...20.1053A}}.
- {{Citation | last1=Chinburg | first1=Ted | last2=Friedman | first2=Eduardo | last3=Jones | first3=Kerry N. | last4=Reid | first4=Alan W. | title=The arithmetic hyperbolic 3-manifold of smallest volume | url=http://www.numdam.org/item?id=ASNSP_2001_4_30_1_1_0 | mr=1882023 | year=2001 | journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV | volume=30 | issue=1 | pages=1–40}}
- {{Citation | last1=Gabai | first1=David | author1-link=David Gabai | last2=Meyerhoff | first2=Robert | last3=Milley | first3=Peter | title=Minimum volume cusped hyperbolic three-manifolds | arxiv=0705.4325 | doi=10.1090/S0894-0347-09-00639-0 | mr=2525782 | year=2009 | journal=Journal of the American Mathematical Society | volume=22 | issue=4 | pages=1157–1215| bibcode=2009JAMS...22.1157G }}
- {{Citation | last1=Matveev | first1=Sergei V. | last2=Fomenko | first2=Aanatoly T. | author2-link=Anatoly Fomenko | title=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 |doi=10.1070/RM1988v043n01ABEH001554 | mr=937017 | year=1988 | journal=Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk | volume=43 | issue=1 | pages=5–22| bibcode=1988RuMaS..43....3M }}
- {{citation|first=Jeffrey|last= Weeks|author-link=Jeffrey Weeks (mathematician)|title=Hyperbolic structures on 3-manifolds|publisher= Princeton University |series= Ph.D. thesis|year= 1985}}