In hyperbolic geometry

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

 obtained by (5, 1) surgery on the figure-8 knot complement. It was introduced by as a possible candidate for the hyperbolic 3-manifold of smallest volume, but the Weeks manifold

Weeks manifold

In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by and 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...

turned out to have slightly smaller volume. It has the second smallest volume

of orientable arithmetic hyperbolic 3-manifolds (where ζ_k is the zeta function of the quartic field of discriminant −283). showed that it is arithmetic.