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A P(1,1,1)

A P(3,-2,-2,2) made by Sandjer, pastry chef at Bretzel Love.


If we call "tangle of order p" the intertwining of two circular helices with |p| crossings, turning counter-clockwise if p > 0 and clockwise otherwise, then the Pretzel link P() is the link obtained by joining tangle of orders , circularly and counter-clockwise.
Tangle of order 5

Tangle of order -5

Pretzel link P(2,-3,4), 
composed of two blades.
Pretzel links can also be characterized by the fact that their associated graph has n vertices placed on a circle with a series of multiple edges of orders (the  edges all have the same sign as ).

For example, the blue graph opposite is associated to P(3,2,2).

The number of blades is equal to the number of even , if at least one of them is even. If all the  are odd, then there are one or two blades depending on whether n is odd or even.

If all the  are positive, then the link has alternating crossings.

The first six prime knots can be seen as Pretzel links with positive coefficients:
Trefoil knot 3.1.1: P(1,1,1) or P(a,b), a+b=3 Figure-eight knot 4.1.1: P(2,1,1) Knot 5.1.1P(1,1,1,1,1) or P(a,b), a+b=5

Knot 5.1.2P(3,1,1) or P(2,1,1,1)
 Stevedore knot 6.1.1: P(2,1,1,1,1) or P(4,1,1)  Knot 6.1.2P(3,2,1)

But this does not hold for the following one (6.1.3): .
For odd values of n, the Pretzel knot P(1,1,...,1) with n "1" is equivalent to the torus knot T(n,2).

The first six prime links can also be obtained as Pretzel links with positive coefficients:
Hopf link 2.2.1: P(1,1) Solomon's knot 4.2.1: P(1,1,1,1) or P(2,2) or P(a,b), a+b = 4 Whitehead link 5.2.1: P(1,2,2)

6.2.3: P(2,1,2,1) or P(2,2,1,1)
6.2.2: P(3,1,1,1)
Solomon 's seal 6.2.1: P(1,1,1,1,1,1) or P(a,b), a+b=6

The link 6.3.2 (Borromean rings) is not a Pretzel link, but 6.3.1 is P(2,2,2): 
When some of the are negative, the minimal number of crossings is less than or equal to the number of visible crossings; for example, the Pretzel link P(2,1,1) opposite is in fact a trefoil knot with 3 crossings, and the link P(2,1,1) is straight out trivial.

The third view displays the link P(2,3,3) that has eight visible crossings and for which 8 is indeed the minimal number of crossings, see the 8.1.19 page.

A real Alsatian Pretzel!

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© Robert FERRÉOL  2018