Torus Knot

Torus Knot
Field: Geometry
Image Created By: 3DXM Consortium
Website: 3DXM Consortium

Torus Knot

In knot theory, a torus knot is a special kind of knot which lies on the surface of an unknotted torus in R3.

Basic Description

In knot theory, a torus knot is a special kind of knot which lies on the surface of an unknotted torus in R3.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Differential Geometry

[Click to view A More Mathematical Explanation]

These knots lie on the following torus of revolution (having ellipses as meridian curves):

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These knots lie on the following torus of revolution (having ellipses as meridian curves):

$x(t) = (aa + bb cos(u)) cos(v)$

$y(t) = (aa + bb cos(u)) sin(v)$

$z(t) = cc sin(u)$

The knots are obtained by putting $u = dd$ $t$, $v = ee$ $t$. The parameters $dd$ and $ee$ should be integers, and the result is referred to as a (dd,ee) knot (The program used to make the image rounds dd and ee before using them.)

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