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