By using LinKnot function fNinvStellar (webMathematica fNinvStellar) we obtained the following number of non-invertible pretzel knots with n crossings (n=10,,17):


n 10 11 12 13 12 15 16 17
1 4 17 51 155 427 1152 2983


The complete consideration of permitted types for non-invertible pretzel knots can be extended to non-invertible pretzel knots with pluses, because every pretzel knot of the form r1,r2,,rt with k pluses can be written as r1,r2,,rt,1,,1 where 1 occurs k times, and ri (i=1,,t, t 3) are R-tangles. The symmetry condition will be applied only to the R-part of the knot, this means, to r1,r2,,rt.
The first non-invertible pretzel knot with pluses is 2 1 1,3,2 1+ with n=11 crossings. For n=12 there are five non-invertible pretzel knots with pluses:

2 2,2 1,3++ 2 1,5,3+ 3 2,2 1,3+
2 2 1,2 1,3+ 2 1 1,2 2,3+

For n=13 there are 22 non-invertible knots:

4 1 1,2 1,3+ 2 1 1,4 1,3+ 2 1 1,2 1,5+

4 1,2 1,3++ 2 2,5,3+ 4 1,2 2,2 1+

2 1 1,2 1,3+++ 2 3,2 1,3++ 3 1 1,2 1,3++ 2 1 1,2 2,2 1++
2 1 3,2 1,3+ 2 3 1,2 1,3+ 2 1 1 2,2 1,3+ 3 1 1 1,2 1,3+
3 2,2 2,3+ 2 2 1,2 2,3+ 2 1 1,2 3,3+ 3 1 1,2 1 1,3+
2 3,2 2,2 1+ 3 1 1,2 2,2 1+ 2 1 1,3 2,2 1+ 2 2 1,2 1 1,2 1+

divided again into three subsets: knots belonging to already derived families, knots generating new families with additional symmetry conditions, and knots generating families without additional conditions for parameters.

In order to obtain families of non-invertible pretzel knots with pluses, we can use the same replacements as in the case of pretzel knots, but also we can replace every sequence of k pluses with a sequence of the same parity. The number of components of a pretzel KL with pluses can be calculated using the same rules as for pretzel KLs.

 

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