Figure-eight knot (mathematics)

The figure-eight knot ( or eight- node ) plays in mathematics, especially in knot theory, a role. He is the mathematical counterpart of Endacht, which is used among other things in sailing.

Parameter representation and invariants

A simple parametric representation of the figure-eight knot is:

The Alexander polynomial of the figure-eight knot is

Its Jones polynomial

The node set of eight node has the presentation

Your Charaktervarietät is the elliptic curve

The A- polynomial is

Properties

The figure eight knot is achiral (also called amphichiral ), that is, it can be deformed into its mirror image. He is not a torus knot.

The figure eight knot is a hyperbolic knot, is its hyperbolic volume

Here is the Bloch -Wigner and Dilogarithm.

The hyperbolic structure is given by the faithful and discrete representation

Cao and Meyerhoff have proved in 2001 that the figure-eight knot, the hyperbolic node is the smallest volume.