Torus knot

A 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:

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

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. ${\displaystyle (r-2)^{2}+z^{2}=1}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle \operatorname {ggT} (p,q)}$

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

${\displaystyle z^{p}+w^{q}=0}$

with the unitary sphere . ${\displaystyle S^{3}\subset \mathbb {C} ^{2}}$

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. ${\displaystyle z^{p}+w^{q}=0}$ ${\displaystyle p:S^{3}-K\rightarrow S^{1}}$ ${\displaystyle p(z,w)={\frac {z^{p}+w^{q}}{\parallel z^{p}+w^{q}\parallel }}}$

Invariants

The number of intersections of a torus knot with is ${\displaystyle (p,q)}$ ${\displaystyle p,q>0}$

${\displaystyle c=\min((p-1)q,(q-1)p).}$

The minimum gender of a Seifert surface of a torus knot with is ${\displaystyle p,q>0}$

${\displaystyle g={\frac {1}{2}}(p-1)(q-1).}$

The Alexander polynomial of a torus knot is

${\displaystyle {\frac {(t^{pq}-1)(t-1)}{(t^{p}-1)(t^{q}-1)}}.}$

The Jones polynomial of a (right-handed) torus knot is

${\displaystyle t^{(p-1)(q-1)/2}{\frac {1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^{2}}}.}$

Web links

Commons : Torus knots and links  - collection of images, videos and audio files

Individual evidence

1. ↑ Torus Knot (s) on MathWorld . Accessed May 22, 2012.
2. ↑ Torus nodes and singularities of complex hypersurfaces