Clover leaf loop

The shamrock loop or trefoil knot is one of the simplest of knots and plays a central role in knot theory . The knot got its name because of its resemblance to shamrocks .

Parameterization and invariants

A simple parametric representation of the clover leaf loop is:

{\ displaystyle x = (2+ \ cos 3t) \ cos 2t, \ qquad y = (2+ \ cos 3t) \ sin 2t, \ qquad z = \ sin 3t.}

The curve defined in this way lies on the torus without overlapping , which is defined in cylindrical coordinates by . This makes the clover leaf loop the simplest example of a torus knot . ${\ displaystyle (r-2) ^ {2} + z ^ {2} = 1}$

The Alexander polynomial of the clover leaf loop is

{\ displaystyle \ Delta (t) = t-1 + t ^ {- 1},}

and its Jones polynomial is

{\ displaystyle V (q) = q ^ {- 1} + q ^ {- 3} -q ^ {- 4}}

or

{\ displaystyle V (q) = q + q ^ {3} -q ^ {4},}

depending on whether she is right or left handed.

The node group has the presentation

{\ displaystyle \ langle x, y \ mid x ^ {2} = y ^ {3} \ rangle \,}

and is therefore isomorphic to the braid group . ${\ displaystyle B_ {3}}$

The knot complement of the clover leaf loop is diffeomorphic to , i.e. the quotient of SL (2, R) after the module group . ${\ displaystyle SL (2, \ mathbb {R}) / SL (2, \ mathbb {Z})}$ ${\ displaystyle SL (2, \ mathbb {Z})}$

symmetry

The clover leaf loop is chiral, i.e. i.e., it is not deformable in its mirror image. Therefore, there are two forms of clover leaf loops that cannot be converted into one another. These are also called right-handed and left-handed shamrocks.

A left-handed clover leaf loop.
A right-handed clover leaf noose.

In art

As a simple knot, the clover leaf loop is often used in the fine arts and iconography . For example, the Valknut and the Triqueta trefoil loops .

The logo of the German Football Association resembles a Valknut in the shape of the clover leaf knot.
A simple Triquetra symbol.
An old Nordic Mjöllnir , pendant with clover leaves.