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TORIC SOLENOID, KNOT AND LINK
one color = one turn around the axis 

Websites:
Wikipedia 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 3_{1} 
T(5,2): pentagram prime knot 5_{1} 
T(7,2): first heptagram prime knot 7_{1} 
T(9,2): first nonagram prime knot 9_{1} 

T(2,5) 
T(2,7) 
T(2,9) 
T(4,3) equivalent to the 19th prime knot with 8 crossings 
T(5,3) equivalent to the 124th prime knot with 10 crossings 
T(7,3) second heptagram 
T(9,4) 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 2_{1}^{2} 
T(4,2), Solomon's knot, prime link 4_{1}^{2} 
T(3,3), prime link
6_{3}^{3}

T(6,2), prime link 6_{1}^{2} 
T(6,3) 
T(6,4), two interlaced trefoil knots 
T(8,2) prime link 8_{1}^{2} 
T(8,4) 
T(8,6) 
T(9,3) 
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 
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, n–1) is equivalent to the nleaved 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