Weeks manifold
Encyclopedia
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 showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by and .
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
:
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
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
obtained by (5, 2) and (5, 1) Dehn surgeries
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
on the Whitehead link
Whitehead link
In knot theory, the Whitehead link, discovered by J.H.C. Whitehead, is one of the most basic links.J.H.C. Whitehead spent much of the 1930s looking for a proof of the Poincaré conjecture...
. It has volume approximately equal to 0.9427... and showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by and .
Since the Weeks manifold is an arithmetic hyperbolic 3-manifold
Arithmetic hyperbolic 3-manifold
In mathematics, an arithmetic hyperbolic 3-manifold is a hyperbolic 3-manifold whose fundamental group is an arithmetic group as a subgroup of PGL...
, its volume can be computed using its arithmetic data and a formula due to A. Borel
Armand Borel
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...
:
where k is the number field generated by θ satisfying θ 3 − θ + 1 = 0 and ζ k is the Dedekind zeta function
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK, is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q...
of k
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.