Weeks manifold

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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 0.9427... and Gabai, Meyerhoff & Milley (2009) showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by Weeks (1985) and 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 A. Borel:

\frac{3 \cdot23^{3/2}\zeta_k(2)}{4\pi^4},

where k is the number field generated by θ satisfying θ 3 − θ + 1 = 0 and ζ k is the Dedekind zeta function of k (Ted Chinburg, Eduardo Friedman & Kerry N. Jones et al. 2001)

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.

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