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Polygram: from the Greek poly "several" and gramma "letter, writing".

A linked polygram is a knot or link obtained by the interlacing of a polygram, and deciding, each time two sides cross, which side passes above the other. There are two remarkable ways to interlace a polygram with Schläfli symbol {p/q}, with p > 2q:
    - either by making each side pass alternatively above and below the sides that are secant (and it can be proved that this is always possible); we get the so-called "alternate" polygram, equivalent to the cylindrical billiard knot or link of type (p,q).
    - or by making each side passing above all the others until its middle, and then below. The resulting knot or link is then equivalent to the toric knot or link T(p,q).
The toric and alternate polygrams are not equivalent when .

If we accept any kind of above and below passages, we obtain all the possible knots and links. Indeed, all knots or links have a projection that is a polygram.

Here are the first examples of toric polygrams, along with the corresponding toric knots:

pentagram {5/2},
or pentacle
prime knot 51


hexagram {6/2}, 
Solomon's seal
or star of David 
prime link 612

heptagram {7/2}
prime knot 71

heptagram {7/3}
prime knot 12 n 242

octogram {8/2}
prime link 812

octogram {8/3}

enneagram {9/2}
prime knot 91

enneagram {9/3}

enneagram {9/4}

decagram {10/2}
prime link 1012


decagram {10/4}

undecagram {11/2}

undecagram {11/3}

undecagram {11/4}

undecagram {11/5}

dodecagram {12/2}

dodecagram {12/3}

dodecagram {12/4}

dodecagram {12/5}


Views showing the trick (due to Alain Esculier) used to trace the polygrams; the strips form in fact a section of hyperboloid.

Here are some examples of "alternate" polygrams (see also the Turk's heads, and the Brunnian links):

Some images of interlaced pentagrams:

Moroccan flag
Ethiopian flag


Some images of interlaced hexagrams:

Israeli flag

A superb hexagram composed of two Penrose triangles.

Composed of two mobius bands


alternate heptagram {7/2}

alternate octogram {8/3}

alternate nonagram {9/3}

alternate decagram {10/2}
medallion of an Iranian Koran, 14th century

dodecagram {12/4}: neither alternate nor toric!

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© Robert FERRÉOL  2018