Torus Knots

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Lexicon ( Torus Knots )

Torus Knots
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Fig. 2.  Illustration of 3D helical motion on a torus surface, and the 2D rectangular transformation of the helical path as parallel lines. From, contributed by the curator of Scholarpedia, Dr. Luigi Chierchia; Dipartimento di Matematica, Universita' di Roma Tre


The torus knot describes the topology of a mathematical relationship of two spin momenta (dual rotations with independent but coupled centers that harmonize at certain integer ratios to form a torus knot.

Equations of a torus knot

About general torus knot lengths

Adapted from: Retrieved 2/11/2019

At any selected point along the length of a torus knot the X, Y, and Z coordinates can be calculated based on the angle to the point around the torus axis.

XYZ coordinates
These point coordinates equations following are accurate only when the torus knot tube windings are equally spaced on a torus surface:


Where p is the number of helical twists around the torus tube, and q is the number of revolutions around the torus axis, and t is the angle of the point from around the torus axis.

Torus knot arc length
The arc length of a minimum torus knot is a calculus equation given as:

L = 0 sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2)dt

Tight approximation of a Torus Knot length

Adapted from:

The usual parametrization of a torus knot is

x(t)=(R+r cos(pt)) cos(qt),
y(t)=(R+r cos(pt)) sin(qt),
z(t)=r sin(pt),

where 0 ≤ t ≤ 2π.

The arc length is

ℓ=∫0,2π p2r2+q2(R+rcos⁡(pt))2dt.
ℓ is the integral from zero to 2π of (p2r2+q2(R+rcos⁡(pt))2) with respect to time.


(R−r)2 ≤ (R+rcos⁡(pt))2 ≤ (R+r)2 for all t,

the arc length satisfies

2π p2r2+q2(R−r)2  ≤  ℓ ≤ 2π p2r2+q2(R+r)2.

Fig. 2. A 7:5 torus knot of a larger, elliptical minor radius as a magnetic-pinch configuration.

Fig. 2 illustrates the simultaneous path of two angular momenta tracing one line they both share at any point in time.

The simple helix around a torus ring is one degree of torus knot, without entanglement, as a degree of X:1, where X is the number twists in the helix during 1 circle around the torus ring.

Fig. 4. A torus knot of twelve rotations around the toroid, and seven helical twists through the toroid hole.

Moebius edge

The torus knot frames the corners of a twisting cycloid through the volume of the torus hosting a knot. The number of corners of the twisting cycloid formed by a torus knot of P:T degree is the value of P. That is, if a torus knot has three poloidal turns, P, for any number of twists through the hole per turn (T), there will yet be three points of the twisting cycloid described by the loops of the torus knot. However, in the fashion of the classical Moebius strip of 1/2 twist, the cycloids of odd-numbered P torus knots has only one corner (1/2, 3/2, 5/2, 7/2, etc.)

The classical Moebius strip (a twisting plane) is a twisting cycloid with an edge describing a torus knot of 2:1 degree. The class of 3:X knots have a triangular cross-section pattern, the 5:X a pentagonal pattern, etc. Notice the square pattern in the cross-section of the 4:3 knot below:

Electromagnetic qualities of torus knot electro-forms

Fig. 4. A 3:5 (p:q) torus knot. The red and blue sections are equal-length halves of the knot in a conjugate symmetry.

The equal length halves of a torus knot (red/blue) as illustrated in Fig. 4 provide two conduction paths for electrical current between the two connector lugs. Current applied to one connector lug will divide between the two conducting paths to combine again at the other lug.

In this dual-path torus knot 3D conduction geometry, there is at least a partial magnetic cancellation for the knot as a whole.

Video illustrations by Daniel Piker

Epicycles on epicycles, Cable knots on cable knots
video by Daniel Piker
An extremely simple system that produces a wide range of curves.
All that is happening is rotation at a constant speed about an axis which is itself rotating about an axis, (rotating about another axis).
There are 4 parameters per arm :
radius, 2 angles for the rotation axis (precession & nutation), and rate of rotation.
When the axes are all parallel the resulting curves are planar and include cardioids, nephroids, epicycloids and many others.
When the second axis is perpendicular to the first, and the rates of rotation are coprime, the curves are torus knots.
When the rates of rotation do not have a common factor the curve is chaotic.
Superhelices and torus knots
A superhelix is simply a helix winding around another helix. First you see the result of increasing the radius, then varying the rate of twist. This causes some surprising topologically complicated knotting.
Next is shown the path of a particle rotated in 4D. By varying the ratio of the 2 components of this double rotation ( ) different torus knots, unknots and links are formed.
Finally, the progression is shown from line to helix to superhelix to 'super-superhelix', a progression that could obviously be carried on recursively.

See Also

A collection of various degrees of torus knots, but illustrated in red and blue halves of equal length paths around the helices of the knot.

Science folders of images