1.11.4 Non-invertible polyhedral knots

The basic polyhedron 6* with two vertices replaced by R-tangles gives knots of the form 6*r1.r2 or 6*r1.r2 0, where none of the R-tangles is of the type . It holds:

 6*r1.r2 = 6*r2.r1,    6*r1.r2 0 = 6*r2.r1 0
From 6*r1.r2 achiral non-invertible knots will be obtained for r1=r2, and chiral non-invertible otherwise. From 6*r1.r2 0 chiral non-invertible knots will be obtained iff r1 ¹ r2.

The next step is to check the same basic polyhedron with three vertices replaced by R-tangles. All knots of the form 6*r1.r2:r3 derived from 6*2.2:2 are non-invertible. Knots of the form 6*r1.r2:r3 derived from 6*2.2:2, and knots of the form 6*r1.r2 0:r3 0 derived from 6*2.2 0:2 0 are non-invertible iff r1 ¹ r3. Knots of the form 6*r1.r2.r3 0 derived from 6*2.2.2 0 are non-invertible iff r2 ¹ r3. Knots of the following forms:

• 6*r1.r2 0.r3 derived from 6*2.2 0.2,
• 6*r1.r2 0::r3 0 derived from 6*2.2 0::2 0, and
• 6*r1.r2.r3 derived from 6*2.2.2
are non-invertible iff they does not contain equal R-tangles. All non-invertible knots mentioned are chiral with the trivial symmetry group.

The results for knots obtained from the basic polyhedron 6* with four R-tangles are following:

• knots of the form 6*r1.r2.r3.r4 derived from 6*2.2.2.2 are achiral non-invertible iff r1 = r4 and r2 = r3, and chiral non-invertible otherwise;
• knots of the form 6*r1.r2:r3.r4 0 derived from 6*2.2:2.2 0 are chiral non-invertible iff r2 ¹ r4, or the tangle types of (r1,r3) are not (,[¥]) or ([¥],), and revertible otherwise;
• knots of the form 6*r1.r2.r3 0.r4 derived from 6*2.2.2 0.2 are chiral non-invertible iff r2 ¹ r3, and revertible otherwise;
• knots of the form 6*r1.r2.r3.r4 0 derived from 6*2.2.2.2 0 are always chiral non-invertible;
• knots of the form 6*r1.r2:r3.r4 derived from 6*2.2:2.2 are invertible achiral iff r1 = r3 and r2 = r4, non-invertible achiral iff r1 = r2 and r3 = r4, revertible iff r1 = r2 or r1 = r4 and r2 = r3, and chiral non-invertible otherwise;
• knots of the form 6*r1.r2 0.r3.r4 0 derived from 6*2.2 0.2.2 0 are chiral non-invertible iff r2 ¹ r3, and revertible otherwise;
• knots of the form 6*r1.r2.r3 0:r4 derived from 6*2.2.2 0:2 are achiral non-invertible iff r1 = r3 and r2 = r4, and chiral non-invertible otherwise;
• knots of the form 6*r1.r2.r3:r4 0 derived from 6*2.2.2:2 0 are chiral non-invertible iff r1 ¹ r3, and revertible otherwise;
• knots of the form 6*r1.r2.r3:r4 0 derived from 6*2.2.2:2 0 are chiral non-invertible iff r1 ¹ r3, and revertible otherwise;
• knots of the form 6*r1.r2 0:r3 0.r4 derived from 6*2.2 0:2 0.2 are revertible iff r1 = r3 and tangle type of (r2,r4) is not (,[¥]) or ([¥],), r2 = r4 and tangle type of (r1,r3) is not (,[¥]) or ([¥],), and chiral non-invertible otherwise.