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TREFOIL KNOT
Noeud de trèfle gauche

Websites:
Knot atlas
etacar.put.poznan.pl/piotr.pieranski/IdealKnots.html

 
Cartesian representation for which the projection on xOy is a hypotrochoid: (right-handed for , left-handed for
Representation by a toric solenoid with coefficient n = 3/2: .
 Other representation: .

See also here an open trefoil knot.

The trefoil knot is the only knot for which the crossing number is minimal, namely 3; there are in fact two of them, they are enantiomorphic (images of one another by reflection).

It is an overhand knot: the ends of which were connected.
 

The trefoil knot is also the torus knot of type (3,2) (3 coils around the torus, on two turns), as well as that of type (2,3):

torus knot with 3 coils on 2 turns

torus knot with 2 coils on 3 turns;
contrary to what it looks like, we get a trefoil knot and not the figure-eight knot.
The trefoil knot is the edge of a Möbius strip with 3 half-twists.

 
 
Conversely, a strip knotted as a trefoil knot yields a Möbius strip with 3 half-twists:
Trefoil knot from a folded strip.
The strip has 6 half-turns, and therefore it is not a Möbius strip (it has two faces, and two edges).
(Logo of Point gamma, student party of Polytechnique).

 
Opposite, a trefoil knot obtained by a closed skew polygonal line with six segment lines; though it might seem, the 7 bars do not have the same lengths (the 3 external bars are shorter).
Anaglyph 3D made by Alain Esculier, to be looked at using red (on the left) and cyan (on the right) glasses.
Opposite, a trefoil knot obtained with 6 segment lines of the same length;
the coordinates of the 6 points a,b,c,d,e,f are:

[[-1, 0, 0], [1, 0, 0], [(13-4*22^(1/2))/25, 6/5, -(4/25)*(-1+22^(1/2))*3^(1/2)], [-1/2, 3/5, (1/2)*3^(1/2)], [1/2, 3/5, -(1/2)*3^(1/2)], [(-13+4*22^(1/2))/25, 6/5, (4/25)*(-1+22^(1/2))*3^(1/2)], [-1, 0, 0]]:


 
It has been proved that it is impossible to obtain a trefoil knot with 5 bars, except in the opposite impossible figure (idea of Alain Esculier, in the style of Oscar Reutersvärd.). For 6 bars, it is also impossible for the segment lines to be pairwise parallel, except again for this impossible figure
Opposite, several views of a trefoil knot composed of polygonal lines with right angles.

It can be proved that this figure represents the smallest trefoil knot passing by the vertices of a cubic lattice.

Two other polygonal versions with 9 and 12 segment lines.

 
The name "trefoil knot" is probably due to the resemblance of the opposite representations to a trefoil leaf:
trèfle gauche
trèfle droit

 
This knot with 4 visible crossings is a trefoil knot (turn the top loop to make the central crossing disappear).

Inverse the two top (or bottom) crossings and you get a figure-eight knot.

The string of this package form a trefoil knot (but, if a crossing is inverted, the knot becomes trivial).

 
We could call "4-leaf trefoil knot" the curve for which the projection on xOy is a hypotrochoid with 4 leaves, parametrized by: . The knot obtained is equivalent to the toric knot of type (4,3), and to the 19th prime knot with 8 crossings. The knot with the same view from above, but with alternate above/below crossings is equivalent to the 18th prime knot with 8 crossings.
This can be generalized to the "n-leaf trefoil knot": which is in fact equivalent to the toric knot of type (n, n – 1). The alternate case would give a Turk's head of type (n, n – 1).

See also the two graphs associated to the trefoil knot, the Seifert surface associated to the trefoil knot, the square and granny knot, sums of two trefoil knots.

The trefoil knot in art:
 
trèfle gauchetrèfle droit

The simple Celtic knot is a trefoil knot

trèfle gauchetrèfle droit
Le noeud de trèfle vu par Escher

The trefoil knot as seen by Escher

sculpture en noeud de trèfle, université de Flensburg
 

Sculpture of a trefoil knot, university of Flensburg
This time, it is a Möbius strip.

The trefoil knot is often used as a logo:
 
Sum of 4 trefoil knots, Celtic on the left, Mongolian on the right.

Behind the logo Woolmark, one can imagine a trefoil knot...

 

Celtic trefoil knot interlaced with a trivial knot.
N° 32 of the links with 9 crossings.

Logo of the Cimat, center for research in mathematics of Guanajuato, Mexico.
The strip has two half-twists, and therefore is not a Möbius strip.


Alien trefoil knot!

 
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© Robert FERRÉOL, Jacques MANDONNET 2018