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RECTANGULAR BILLIARD KNOT AND LINK
The " (6,2) "
Studied by Jones
and Przytycki in 1998.
Links: undergraduate research work on billiard trajectories Constructing the (5,3) by hand! Images made by Alain Esculier. 
A rectangular billiard knot is the knot
obtained from the closed trajectory of a ball on a billiard with rectangular
edge, by modifying the crossing points in alternate above/below passages.
If the dimensions of the billiard are L,L' and
the ball starts rolling from a side with length L (and not in the
corner) following a trajectory forming a slope a with respect to
this side, then the trajectory is closed iff the ratio L/L' over
a
is a rational number p/q (with p and q coprime).
In this case, the ball bounces p times on the sides with length L and q times on the sides with length L'. All in all, there are crossings. 
Here p = 5 and q =3; 5 bounces for each horizontal side,
3 for each vertical side,

Except the cases for which they start at a corner, the curves have the same topology, and therefore yield a unique knot, that we call of type (p, q). 

Up to a scaling, we can suppose
 that the billiard is a square (and the slope is equal to p/q),  or that L/L'=p/q, in which case the slope is equal to one (the crossings form right angles). 

The billiard knot of type (p,q) can also be obtained
from the planar
Lissajous curve.
Indeed, if c is an even, continuous function decreasing on and such that c(0)=1 and , then the curve gives, for , a billiard knot of type (p,q) (p > q) inscribed in a rectangle with sides p and q (for , we arrive at the corners). For , we get a Lissajous curve, and for , a billiard trajectory. For , the bounces are evenly spaced on the sides. 
case p = 4, q = 3 
The above/below alternation can be obtained by a 3D Lissajous
curve, or, which is equivalent, by the trajectory of a ball (not subject
to gravitation) in a parallelepipedic billiard.
Equation with the above notations: . See at Lissajous knot. 

Examples:
q = 1: the knot is trivial  p = 3, q = 2: we get the fourth prime knot with 7 crossings.  p = 4, q = 3, prime knot with 17 crossings.  p = 5, q = 3 




Artistic productions: Buddhist, Islamic, Celtic, Roman, or maritime art! 
Picture taken in Kathmandu: B. Ferreol. 
Image taken from the very interesting blog Nicomatelotage. 
Celtic knot 
Variations and generalizations:
1) For p and q non coprime, if we trace all the trajectories with p evenly spaced bounces on two opposite sides, and q bounces on the two other sides, then we get a link with gcd(p,q) components.
Examples:
p = 2, q = 2: we get Solomon's knot, the simplest non trivial link  p = 3, q = 3: link with 12 crossings and 3 components, indexed by 12x379 in knotilus. Sometimes called triple Solomon knot.  p = 4, q = 2: link with 10 crossings and 2 components, indexed by 101 in knotatlas.  p = 4, q = 4.
Quadruple Solomon knot. 
p = 8, q = 8 





Roman mosaic 
Islamic link (Marrakech) 
Mongolian pattern, that can be found as a decoration in yurts 
Roman mosaic (villa casale) 
The graph of the Turk's head of type (p,q) (coprime or not) is a rectangular grid of p–1 times q–1 squares; opposite, the case (5,3). 
2) We can also consider the curves obtained when the ball starts from a corner. In this case, the curve is open, but can be closed in various ways; and we can superimpose several curves.
Examples in the case (3,2):
Closing only one open curve creates a trivial knot, but
the superimposition of two open curves is interesting.
It is the Carrick
bend which leads, after closing, either to the 18th
prime knot with 8 crossings : 8.1.18, or to the 7th
prime link with 8 crossings and two loops: 8.2.7.




Knot 8.1.18 

Examples in the case (4,3):
If the curve is closed, we get the trefoil knot.  If two open curves are superimposed and the different blades connected, we get a knot with 18 crossings indexed by 18x1230179 in knotilus; it is used in the fabrication of doomarts.  If the identical blades are connected, we get a link with 18 crossings indexed by 18x2  410219 in knotilus. 






Example in the case (7,3):
Example in the case (1,1); closing gives the Whitehead link.  Example in the case (3,3): 


3) We can also consider non alternate above/below crossings.
We get knots for which the minimal crossing number is
less than the number of crossings of the curve.



Above, two crossings can be unfolded (topright): we get the knot 5.1.2.  Here, 4 crossings can be unfolded, we get the trefoil knot.  This knot was obtained by following the billiard ball with a passage below when we cross a previous line. Therefore, we can unfold starting from the end. Since this works every time, there always exists a configuration yielding the trivial knot. 
4) The rectangular billiard can be replaced by a convex
polygonal billiard.
With any type of crossings, we can get all the possible
knots, even by only considering billiards the edge of which is a regular
polygon. Indeed, every knot
has a projection that is a crossed regular polygon. See also the polygram
knots.
With a triangular billiard, for example, we get the trefoil knot: 


4 loops link got with a cruciform billiard.
On the right, variant with an additional loop, idea of Alain Esculier. 


This roman mosaik from the GalloRoman villa of Seviac represents an interlacing which is found to be obtained by the routes of two balls in a cruciform billiard. 

See also the cylindrical billiard knots, or Turk's heads, the linear Celtic knots.
Frontispiece of the chapel of Murato in Corsica: it is a (22,3).
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© Robert FERRÉOL 2018