hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

, the Meyerhoff manifold is the

arithmetic hyperbolic 3-manifold In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arith ...

obtained by

(5,1)

surgery Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pat ...

on the figure-8

knot complement In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...

. It was introduced by as a possible candidate for the hyperbolic 3-manifold of smallest volume, but the

Weeks manifold 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 0.942 ...

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

V_m = 12\cdot(283)^\zeta_k(2)(2\pi)^ = 0.981368\dots

of orientable arithmetic hyperbolic 3-manifolds, where

\zeta_k

is the

zeta function In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * A ...

of the quartic field of discriminant

-283

. Alternatively, :

V_m = \Im(\rm_2(\theta)+\ln, \theta, \ln(1-\theta)) = 0.981368\dots

where

\rm_n

is the

polylogarithm In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...

and

, x,

is the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

of the complex root

\theta

(with positive imaginary part) of the quartic

\theta^4+\theta-1=0

. showed that this manifold is arithmetic.

References

* * * 3-manifolds Hyperbolic geometry {{hyperbolic-geometry-stub

See also

References