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… () and txt showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by txt as well as txt.

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 \cdot23^{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. [1] Alternatively,

V_w = \Im(\rm{Li}_2(\theta)+\ln|\theta|\ln(1-\theta)) = 0.942707\dots

where \rm{Li}_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. [2]

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

References

  1. ^
  2. ^ Mednykh, Alexander & Vesnin, Andrei (1998). "Visualization of the isometry group action on the Fomenko–Matveev–Weeks manifold". Journal of Lie Theory. 8 (1): 51–66. Heldermann Verlag. Retrieved 2025-05-28.