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one color = one turn around the axis
Knot atlas

Cylindrical equation:  (n > 0)
Cartesian parametrization: .
Curvilinear abscissa: .

The toric solenoids are the solenoids the central curve of which is a circle; therefore, they coil evenly around a torus. They can also be seen as the trajectory of a point with a uniform circular motion in a plane turning uniformly around an axis.
The toric solenoids are also obtained as the intersection between the generalized Plücker's conoid and the torus with center O and major and minor radii R and r. Opposite, the cases n = 2 and 3: the intersection is composed of several rotated solenoids.

When the torus is reduced to a sphere (R = 0), we get the clelias.
The projections on xOy are the conchoids of roses.

spindle torus

View from above: conchoid of a rose

open torus

View from above: conchoid of a rose

When n is a rational number p/q, and R > r, the curve is closed and simple, and the knot associated to the corresponding toric solenoid is the torus knot T(p, q), that has p coils around the torus and q turns around the axis, and always is a prime knot. The knots T(p, q) and T(q, p) are equivalent (to go from (p,q) to (q,p), pass a needle in the bore of the torus).

Every knot that has a representation without crossings on the torus is a torus knot of this type.
Every right section of the tube has q blades and the view from above shows p (q 1) crossings; it was proved that for p > q, this number of crossings is the minimal number of crossings of the corresponding knot, (the latter is therefore equal to q (p 1) for p < q).
For n = 1 (and also for any n integer or reciprocal of an integer), we get the trivial knot (but contrary to what might be expected, the solenoid is not a Villarceau circle of the torus).

For q = 2 (respectively p = 2), we get knots with p (respectively q) crossings:

T(3,2): trefoil knot
prime knot 31

T(5,2): pentagram
prime knot 51

T(7,2): first heptagram
prime knot 71

T(9,2): first nonagram
prime knot 91
T(2,3): we get the trefoil knot and not the figure-eight knot, contrary to appearances (for the figure-eight knot, the crossing on the left should be inverted).





equivalent to the 19th prime knot with 8 crossings

equivalent to the 124th prime knot with 10 crossings

second heptagram

third nonagram

The toric solenoids for q = 2 are edges of Möbius strips with p torsions:

n = 1/2: edge of the classic Möbius strip (one torsion)

n = 3/2: edge of the Möbius strip with 3 torsions

When p and q are not coprime, if we write d = gcd(p, q), p' = p/d, q' = q/d, n = p/q = p'/q', the toric solenoid of type (p', q') and its d 1 images by consecutive rotations of angle  around the axis of the torus form a link of d torus knots of type (p', q'), called torus link T(p, q). It is also a prime link with minimal number of crossings p (q 1) for p > q (?).

Here are some examples:

T (2,2), 
Hopf link,
prime link 212

Solomon's knot,
prime link 412

T(3,3), prime link 633
(fake Borromean rings)

T(6,2), prime link 612


T(6,4), two interlaced trefoil knots

prime link 812




We can make the torus link T(p, q) by placing q blades of the same length side by side and applying a torsion of p/q turns and glueing the blades at their ends.

For example, for the knot (8, 3), there are three glued blades after a torsion of 8/3 turns

The same knot in a sculpture by J. Robinson
Philip Trust Collection
See magnificent figures and explanations on the website:

The torus knots are also sometimes defined on the Clifford torus; their parametrization is much simpler: ; by identification of  and , they can also be seen as the image of the unit circle  by the map: .

The torus knots and links for p > 2q are equivalent to the polygram knots and links.
They are also the "edges" of the rotoidal prisms.
The torus knot T(n, n1) is equivalent to the n-leaved trefoil knot.
Compare to the Turk's heads, that have the same view from above, but with alternate crossings.

Also compare to the geodesics of the torus.

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© Robert FERRÉOL  2018