Basic graph theory

1.11.2 Non-invertible pretzel knots

Every alternating pretzel KL is of the form r₁,r₂,¼,r_i, where r_i are R-tangles (i ³ 3). Pretzel knots are obtained as the numerator closures of the pretzel tangles of the reduced type [1], or [¥]. Hence, non-reduced types of pretzel KLs can be:

[0,¼,0,1,¼1], where 1 has an odd number, and 0 an arbitrary number of occurrences;
[0,¼,0,1,¼1,¥].

The main question is wether it is possible to determine non-invertibility of pretzel knots according to tangle types. Every pretzel knot can be drawn as a regular t-gon with vertices denoting R-tangles, called t-diagram. In a t-diagram vertices by themselves are treated as symmetrical, and the mirror line contains at least one vertex.

Conjecture (Non-invertibility criterion for pretzel knots) A pretzel knot is non-invertible iff its type symbol consists only from 0-s and 1-s, and its t-diagram is not mirror-symmetrical.

We give detailed description of pretzel knots with n £ 5:

for n=3 allowed types are [1,1,1], [0,0,1], and all three R-tangles must be mutually different;
for n=4 allowed types are [1,1,1,0], [0,0,0,1], and from the symmetry condition follows that the R-tangles at the first and third position must be different;
for n=5 allowed types are [1,1,1,1,1], [0,0,1,1,1], [0,0,0,0,1], and the symmetry condition can be easily recognized from a t-diagram.

For example, the pretzel knots 3,3,3,5,7 and 3,3,5,3,7 are non-invertible, while 3,5,5,3,7 is invertible, and all have the same type [1,1,1,1,1].

In general, the necessary condition for invertibility of pretzel knots is that the sum of numbers in knot type must be odd. Sufficiency is determined, based on symmetry condition, from the t-diagram.