Torus knot

A torus knot is a knot in knot theory, which can be drawn on a ( unknotted ) torus in three-dimensional space.

Parameterization

A torus knot is represented by two integers, prime parameter ( p and q), which specify how often the node the torus "around" and " through the hole of the torus " orbits. A parametric representation of a Torusknotens with parameters p and q is:

The curve is zero overlap on the torus, which can be defined in cylindrical coordinates by. This one really gets a torus knot here, and must be relatively prime, otherwise you get a tangle with components.

Properties

The simplest non-trivial torus knot is the trefoil knot. A torus knot if and only trivial if p = ± 1 or q = ± 1 Each ( non-trivial ) torus knot is chiral, that is, he is not in his reflection deformable.

The complement of a Seifert fibration is a Torusknotens. In particular, torus knots are not hyperbolic knots.

Torus knots arise in the singularity theory as a section of the complex hypersurface

With the unit sphere.

The complement of the Torusknotens is a fiber bundle over the circle with monodromy of finite order. If the node is given as the intersection of the unit sphere with the hypersurface, can be defined by the fibrillation.

Invariants

The intersection number of a (p, q) with Torusknotens p, q > 0,

The minimal genus of a Seifert surface of a Torusknotens with p, q > 0

The Alexander polynomial of a Torusknotens is

Is the Jones polynomial of a ( right-handed ) Torusknotens