By using LinKnot function fNinvStellar
we obtained the
following number of non-invertible pretzel knots with n
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,r2,¼,rt with k pluses can be written as r1,r2,¼,rt,1,¼,1 where 1 occurs k times, and ri (i=1,¼,t, t ³ 3) are R-tangles. The symmetry condition will be applied only to the R-part of the knot, this means, to r1,r2,¼,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:
For n=13 there are 22 non-invertible knots:
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.