Torus knot
A torus knot 's knot theory , which on one (unknotted) a node torus can be drawn in three-dimensional space.
Parameterization
A torus knot is determined by two integer, relatively prime parameters ( p and q ), which indicate how often the knot circles the torus around the outside and through the hole . A parametric representation of a torus knot with parameters p and q is:
The curve lies on the torus without overlapping, which can be defined in cylindrical coordinates by . So you really here gets a torus knots, have and be relatively prime, otherwise we obtain a entanglement with components.
properties
The simplest non-trivial torus knot is the clover leaf loop . A torus knot is trivial if and only if p = ± 1 or q = ± 1. Every (non-trivial) torus knot is chiral, that is, it cannot be deformed into its mirror image.
The complement of a torus knot is a Seifert fiber . In particular, torus nodes are not hyperbolic nodes .
In the singularity theory, torus knots arise as intersections of the complex hypersurface
with the unitary sphere .
The complement of the torus knot is a fiber bundle over the circle with monodromy of finite order. If the knot is given as the intersection of the unit sphere with the hypersurface , the grain can be defined by.
Invariants
The number of intersections of a torus knot with is
The minimum gender of a Seifert surface of a torus knot with is
The Alexander polynomial of a torus knot is
The Jones polynomial of a (right-handed) torus knot is
Web links
Individual evidence
- ↑ Torus Knot (s) on MathWorld . Accessed May 22, 2012.
- ↑ Torus nodes and singularities of complex hypersurfaces