next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

KNOT


Prime knot with 19 crossings

A knot, in the mathematical sense, can be defined as an equivalence class of smooth curves of  closed without self-intersection, two curves being equivalent if one can be transformed into the other in  continuously, the curve remaining closed and without self-intersection throughout the transformation.
The crossing number of a knot is the minimal number of double points of the planar projections of its representations that do not have points of order greater than or equal to 3. The knot that has a representation without crossings is called trivial knot, or unknot.
 
The sum of two knots A and B is defined as the knot obtained by cutting A and B, calling the four ends A1, A2, B1, B2 and glueing A1 to B1, and A2 to B2 (the resulting knot does not depend on where the cuts were made). A prime knot is a knot that cannot be the sum of two non trivial knots. Other knots (except the trivial one) are called composite knots, and can be written in a unique way as the sum of prime knots.

As the diagram on the right shows, this knot can be decomposed into 
a figure-eight knot (4 crossings) and a prime knot with 7 crossings 7.1.7.

Here are the diagrams of the prime knots with from 0 to 9 crossings:

(see also this link)

The first six are Pretzel knots.
See on this page the graph associated to the knot.
See the trefoil knot 3.1.1, the figure-eight knot 4.1.1, the square and granny knots, the Carrick bend, the stevedore knot 6.1.1, the toric knots, the polygram knots, the rectangular or cylindrical billiard knots, the polygonal knots, the Lissajous and 3D billiard knots, the linear Celtic knots.
See also the conchoids of roses that provide many knots with rotation symmetry.

Finally, see the links, as well as the Seifert surfaces, surfaces the edges of which are a knot.
Compare to the generic curves.
 
 
Impossible knot by Oscar Reutersvärd.

It is a prime knot of type 8.1.16.

Islamic knot; 16 crossings.
Gauss code:
1,2,3,4,5,6,7,8,9,10,11,12,2,13,4,14,6,15,8,16,10,1,12,3,13,5,14,7,15,9,16,11

Other links on knots:
Book by Peter Cromwell
Knot-atlas
Java applet to determine the Gauss code of a knot from its drawing: knotilus.math.uwo.ca/javasketch.php
Website that allows to recover a prime knot from its Gauss code: knotilus.math.uwo.ca
www.earlham.edu/~peters/knotlink.htm
www.ulb.ac.be/soco/matsch/recherche/11/noeuds/noeuds04.htm
www.knotplot.com/download to download Robert Charein's magnificent software: "knotplot".
 
 
next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL   2018