Table of known symmetries

(See index page for description of the symmetries D2(r), D4, D6, D8, D10, D14, I2, I8, Z2, Z4, Z6 listed in the first five columns of data, p 336 of Burde G. & Zieschang H., Knots, Walter de Gruyter, 2003, for the periods in the penultimate column, and FSG for a note on the "full symmetry group" of the last column.)

D2(r) D2k Z2k I reversibleamphichiral periods FSG
31 D4, D6 2, 3 Z2
41 D4 Z4 I8 2 D8
51 D4, D10 2, 5 Z2
52, 61, 62 D4 2 D4
63 D4 Z4 2 D8
71 D4, D14 2, 7 Z2
72, 73 D4 2 D4
74 D4 2 D8
75, 76 D4 2 D4
77 D4 2 D8
81, 82 D4 2 D4
83 D4 Z4 I8 2 D8
84, 85, 86, 87, 88 D4 2 D4
89 D4 I4 2 D8
810 none D2
811 D4 2 D4
812 D4 Z4 2 D8
813, 814, 815 D4 2 D4
816 none D2
817 none D2
818 D4, D8 Z8 2, 4 D16
819 D4, D6, D8 2, 3, 4 Z2
820 none D2
821 D4 2 D4
12a1202 Z2, Z6 D12
15331 Z2
D2(r) D2k Z2k I reversible amphichiral periods FSG

Directly from the definitions in the index page, we see that any D2k symmetry, for k ≥ 2, implies D2(r) symmetry, and if k is even it implies period 2; also, of course, it implies D2j symmetry for any factor j of k. Any Z2k symmetry implies positive amphichirality, and D2(r) symmetry implies reversibility. I2 symmetry implies negative amphichirality. Any D2k symmetry, for k ≥ 2, implies that k is a period, while a knot has D2(p) symmetry iff 2 is a period. Proofs of asymmetry on the classification here seem to be hard. I am relying on Knotinfo to tell me which knots are reversible (other than those for which I can exhibit a D2(r) symmetry) and which are amphichiral (other than those for which I can exhibit a Z2k symmetry). Moreover, I have no method of finding these symmetries other than fiddling with string and hopeful guesswork. So there may well be surprises; in particular, I came on some of the D4 symmetries by chance, and conceivably there are more. I have listed some obvious questions.

The "Full Symmetry Group" listed in Knotinfo does not include the same information as the symmetries discussed here (for instance, it ignores periodicity), but is clearly related. For instance, I note that 810, 816, 817 and 820, which are the prime knots of 8 or fewer crossings for which 2 is not a period, are also the knots of 8 or fewer crossings which have "full symmetry group D2".

I found 12a_1202 and 15331 in the course of searching for knots with Z2 symmetry; the first is based on a Hamiltonian path along the edges of an octahedron, and the second on a randomly generated Fourier series. I do not have a definite crossing number for 15331, and I do not know what is the least crossing number of any non-trivial knot with Z2 symmetry.

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