1. Volume

2. Related manifolds

3. References

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 0.94270736… ({{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}}

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.

References

• {{citation

• {{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|authorlink=Jeffrey Weeks (mathematician)|title=Hyperbolic structures on 3-manifolds|publisher= Princeton University |series= Ph.D. thesis|year= 1985}}

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