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	reversible	amphichiral	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 D_2k symmetry, for k ≥ 2, implies D₂(r) symmetry, and if k is even it implies period 2; also, of course, it implies D_2j symmetry for any factor j of k. Any Z_2k symmetry implies positive amphichirality, and D₂(r) symmetry implies reversibility. I₂ symmetry implies negative amphichirality. Any D_2k 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 Z_2k 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.

List of knots, questions, home page.

6.7.15