Weeks manifold

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:

: $V\_w\; =\; \backslash frac\{3\; \backslash cdot23^\{3/2\}\backslash zeta\_k(2)\}\{4\backslash pi^4\}\; =\; 0.942707\backslash dots$

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

: $V\_w\; =\; \backslash Im(\backslash rm\{Li\}\_2(\backslash theta)+\backslash ln|\backslash theta|\backslash ln(1-\backslash theta))\; =\; 0.942707\backslash dots$

where $\backslash rm\{Li\}\_n$ is the polylogarithm and $|x|$ is the absolute value of the complex root $\backslash theta$ (with positive imaginary part) of the cubic.

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. {{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

}}

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.

• Meyerhoff manifold – second small volume

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Category:3-manifolds

Category:Hyperbolic geometry