the simplest manifold

Here we introduce the simplest self-persistent stage in topology—the hyperbolic figure eight knot. This topology is defined as a double-cover of the simplest possible 3-manifold ( the Gieseking Manifold ) defining the complement with the smallest possible volume.

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Videos by Jeff Chapple, based on work by Henry Segerman.

The constructive equation for the hyperbolic figure eight knot volume is:

Equation 1

Where Li_2(x) = the polylogarithm of order 2 (the dilogarithm ), i = the imaginary unit, phi_i = the imaginary golden ratio, V_fe=2G_Gi, and G_Gi = Gieseking's constant for the minimum 3-manifold.

Equation 5aEquation 5b

The real conjugate of the hyperblic figure eight knot defines a double covered sphere, divided into 288 pieces—each associated to a unique constant of Nature.

Equation 2

Where pi = Archimedes' constant.

These constructions pull apart into two dilogarithm pieces, made of identical real and inverse imagainry parts.

Equation 5aEquation 5b

Where gamma(x) = the gamma function.

Switching the arguments of that construction, from the imaginary golden ratio to the imaginary unit, yields twice Catalan's constantK.

Equation 2

These constructions naturally divide the world up into electron, proton, and neutron radii.

Equation 5aEquation 5b

Where r_n = the neutron radius, gamma(x) = the classical electron radius, and r_+ = the proton rms charge radius.