Basic graph theory

By using LinKnot function fNinvStellar (webMathematica fNinvStellar) we obtained the following number of non-invertible pretzel knots with n crossings (n=10,¼,17):

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,r₂,¼,r_t with k pluses can be written as r₁,r₂,¼,r_t,1,¼,1 where 1 occurs k times, and r_i (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,r_t.

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.