Unknot

The trivial node (also unknot ) is the simplest mathematical nodes, a simple closed loop that is not tied (that can be pulled apart without cuts a smooth ring). It plays a role in knot theory.

Many occur in practice node, for example, the trumpet node and the Würgeknoten are trivial knots.

Node Theoretical properties

A curve representing the trivial knot, for example,

A node is a trivial node if ( any of its " cut the cord " ) can be converted into the above curve by a continuous deformation. There are quite complicated -looking nodes that are trivial in reality, an example shows the image on the right.

The Jones polynomial of the trivial knot is:

Its Alexander polynomial is also equal to 1

A knot K in the 3- sphere if and only trivial if the complement is homeomorphic to Volltorus is.

1961 developed the mathematician Wolfgang hook an algorithm with which one can determine whether a node diagram shows a trivial knot or not. Therefore, he used Seifert surfaces. With hook algorithm can generally decide whether two hook -manifolds are homeomorphic. ( Hook -manifolds are irreducible 3-manifolds containing an incompressible surface - in the case of Knotenkomplements the Seifert surface is incompressible this area. )

Hatred / Lagarian / Pippenger proved in 1999 that Unverknotetsein in the complexity class NP, ie a " certificate" that a node is trivial, can be verified in polynomial time.

Under the assumption that the generalized Riemann conjecture is correct, Greg Kuperberg proved in 2011 that also Verknotetsein in NP.

It is not known whether one can discover the trivial knot with the Jones polynomial, ie whether valid only for the trivial knot. However, this makes the Heegaard Floer homology.