Till now, the symmetry group of all chiral non-invertible knots was trivial, but chiral non-invertible knots of the form 6*r1.r2 0:r3 0.r4 with r1 = r3 or r2 = r4 have the symmetry group Z2. Knots of the form 6*r1.r2:r3 0.r4 0 derived from 6*2.2:2 0.2 0 are revertible iff r1 = r3 and tangle type of (r2,r4) is not ([0],[]) or ([],[0]), r2 = r4 and tangle type of (r1,r3) is not ([0],[]) or ([],[0]), non-invertible achiral iff r1 = r2 and r3 = r4, and chiral otherwise. Among the chiral knots of this form, knots with r1 = r3 or r2 = r4 have the symmetry group Z2, and a trivial symmetry group otherwise. Knots of the form 6*r1.r2.r3:r4 derived from 6*2.2.2:2 are chiral non-invertible iff r1 r3, and revertible otherwise.

There is an interesting connection between the knots derived from 6*2.2:2 0 and 6*2.2 0:2 0. For every n they generate the same number of knots, and the derived knots from those two classes mutually correspond with regard to all kinds of symmetries and invertibility. The same holds for the following pairs: 6*2.2 0::2 0 and 6*2.2.2, 6*2.2.2.2 and 6*2.2.2 0:2, 6*2.2 0.2.2 0 and 6*2.2.2:2 0, 6*2.2.2 0.2 and 6*2.2.2:2

The similar connection exists between 6*2.2 0:2 0.2 and 6*2.2:2 0.2 0 if we consider only the number of knots derived, the number of chiral non-invertible knots among them, and their distribution according to the symmetry groups (Z2 or trivial).

In the same way it is possible to find non-invertibility criteria for all knots derived from different basic polyhedra replacing vertices by R-tangles.

Next class of non-invertible polyhedral knots will be obtained by replacing vertices of a basic polyhedron by pretzel (stellar) tangles. For example, all knots of the form 6*p1.r1, where p1 is the pretzel tangle of the pretzel type [], and r1 is R-tangle of the type [0], are chiral non-invertible. We already mentioned that a pretzel tangle p1 will be of the type [] if it consists of an arbitrary number of R-tangles of the types [1], [], and from exactly one R-tangle of the type [0]. First examples are 11-crossing chiral non-invertible knots 6*(3,2).2, 6*(2,3).2, 6*(2 1,2).2, and 6*(2,2 1).2. We conclude that all knots derived from the basic polyhedron 6* with one pretzel tangle p1 and one R-tangle r1 will be chiral non-invertible.

The first class of polyhedral knots with pretzel tangles that requires symmetry discussion are knots of the form 6*p1.p2, where p1, p2 are pretzel tangles of the types ([],[]), or ([0],[0]), respectively. If p1, p2 are pretzel tangles of different types, [0] and [], the corresponding knots are chiral non-invertible. Knots of the form 6*p1.p2 where p1, p2 are pretzel tangles of the types ([], []) are achiral non-invertible iff p1, p2 are mutually palindromic pretzel tangles, and chiral non-invertible otherwise. For example, the knot 6*(3,2).(2,3) is achiral non-invertible, and 6*(3,2).(3,2) is chiral non-invertible. The same criterion holds for the knots of the same form 6*p1.p2, where p1, p2 are pretzel tangles of the types ([0],[0]).

In the same way we can conclude that all knots of the form 6*p1.r1 0, where p1 is a pretzel tangle, and r1 is R-tangle, are chiral non-invertible.

Knots of the form 6*p1.p2 0, where p1, p2 are pretzel tangles of different types [0], [], are chiral non-invertible. If the pretzel tangles p1, p2 are of the same type ([0],[0]), or ([],[]), knots of the form 6*p1.p2 0 are chiral non-invertible iff p1 p2. For example, knot 6*(3,2).(3,2) 0 is invertible, and 6*(5,2).(3,2) 0 is chiral non-invertible.

In general, for every basic polyhedron infinite classes of non-invertible knots derived from it can be recognized.

Replacement of the tangle of particular reduced type ([0k], [1k], or [k]) by a tangle of the same type preserves the number of components. Hence we have the following main conjecture:

Conjecture Given a non-invertible knot K, every replacement of a tangle of the specific type by a tangle of the same type, respecting symmetry conditions, gives a non-invertible knot and preserves chirality.

In other words: non-invertibility is type-dependent and symmetry-dependent property, and not only the property of particular knots.

 

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