Trefoil knot

The trefoil knot or the trefoil knot is one of the simplest nodes and plays a central role in knot theory. The node has its name because of its similarity to shamrocks.

Parameterization and invariants

A simple parametric representation of the trefoil knot is:

The so- defined curve is zero overlap on the torus, which is defined in cylindrical coordinates by. For the trefoil knot is the simplest example of a Torusknotens.

The Alexander polynomial of the trefoil knot is

And its Jones polynomial is

Depending on whether it is right - or left-handed.

The node group has the presentation

And thus is isomorphic to the modular group.

Symmetry

The trefoil knot is chiral, ie they are not in her reflection deformable. Therefore, there are two non- interconvertible forms of cloverleaf loops. These are also called right-handed and left-handed trefoil knot.

A right-handed trefoil knot.

In the arts

As a simple knot, the trefoil knot is often found in fine art and iconography. Thus, for example, the Valknut and Triqueta cloverleaf loops.

A Triquetra.