ウィークス多様体のソースを表示

{{要改訳}}
数学において、'''ウィークス多様体'''(Weeks manifold)（'''フォメンコ・マットヴェーエフ・ウィークス多様体'''(Fomenko–Matveev–Weeks manifold)と呼ばれるときもある）は、{{仮リンク|ホワイトヘッドリンク|en|Whitehead link}}(Whitehead link)上の (5,&nbsp;2) と (5,&nbsp;1) の[[デーン手術]]によって得られる閉じた[[双曲3次元多様体]]である。ウィークス多様体は、約 0.9427... に近い体積を持ち、{{harvtxt|Gabai|Meyerhoff|Milley|2009}} により、閉じた向き付け可能な双曲3次元多様体の最小の体積であることが示された。この多様体は、独立に、{{harvtxt|Weeks|1985}} と {{harvtxt|Matveev|Fomenko|1988}} により発見された。 

ウィークス多様体は[[数論的双曲3次元多様体]]であるので、その体積は数論的なデータを使い計算することができ、[[アルマン・ボレル]]は次の公式を与えた。

: <math>\frac{3 \cdot23^{3/2}\zeta_k(2)}{4\pi^4},</math>

ここに、''k'' は ''θ''<sup>&nbsp;3</sup>&nbsp;&minus;&nbsp;''θ''&nbsp;+&nbsp;1&nbsp;=&nbsp;0 を満す ''θ'' により生成される[[代数体|数体]]であり、''ζ''<sub>&nbsp;''k''</sub> は ''k'' の[[デデキントゼータ函数]]である{{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}}。

ホワイトヘッドリンク上のカスプをもつ (5,&nbsp;1) デーン手術により得られる双曲 3-次元多様体は、[[8の字結び目]]の[[結び目補空間]]の兄弟のような多様体である。8の字結び目の結び目補空間とその兄弟の多様体は、任意の向き付け可能なカスプを持つ双曲3次元多様体の中で最小の体積を持つ。このように、ウィークス多様体は、2つの最小の体積を持つ向き付け可能なカスプを持つ双曲3次元多様体の双曲デーン手術により得ることができる。
<!--In [[mathematics]], the '''Weeks manifold''', sometimes called the '''Fomenko–Matveev–Weeks manifold''', is a closed [[hyperbolic 3-manifold]] obtained by (5,&nbsp;2) and (5,&nbsp;1) [[Dehn surgery|Dehn surgeries]] on the [[Whitehead link]].  It has volume approximately equal to 0.9427... and {{harvtxt|Gabai|Meyerhoff|Milley|2009}} showed that it has the smallest  volume of any closed orientable hyperbolic 3-manifold.  The manifold was independently discovered by {{harvtxt|Weeks|1985}} and {{harvtxt|Matveev|Fomenko|1988}}. 

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|A. Borel]]:

: <math>\frac{3 \cdot23^{3/2}\zeta_k(2)}{4\pi^4},</math>

where ''k'' is the [[number field]] generated by ''θ'' satisfying ''θ''<sup>&nbsp;3</sup>&nbsp;&minus;&nbsp;''θ''&nbsp;+&nbsp;1&nbsp;=&nbsp;0 and ''ζ''<sub>&nbsp;''k''</sub> is the [[Dedekind zeta function]] of&nbsp;''k'' {{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}}

The cusped hyperbolic 3-manifold obtained by (5,&nbsp;1) Dehn surgery on the Whitehead link is the so-called sibling manifold, or sister, of the [[figure-eight knot (mathematics)|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.-->

==参考文献==
*{{citation
 | last1 = Agol | first1 = Ian
 | 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}}.
*{{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}}
*{{Citation | last1=Matveev | first1=S. V. | last2=Fomenko | first2=A. 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}}
*{{citation|first=Jeffrey|last= Weeks|authorlink=Jeffrey Weeks (mathematician)|title=Hyperbolic structures on 3-manifolds|publisher= Princeton Univ.|series= Ph.D. thesis|year= 1985}}

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[[Category:3次元多様体]]
[[Category:双曲幾何学]]
[[Category:数学に関する記事]]