Knot Symmetry

A knot symmetry is a homeomorphism of the pair of spaces , where is a knot. The symmetry group consists of equivalence classes of these maps under continuous families of such homeomorphisms (Hoste et al. 1998). Hoste et al. (1998) consider four types based on whether a symmetry preserves or reverses the orientations of and ,

1. preserves , preserves (including the identity map),

2. preserves , reverses ,

3. reverses , preserves ,

4. reverses , reverses .

This then gives the five possible classes of symmetry summarized in the table below.

class	symmetries	knot symmetries
	1	chiral, noninvertible
	1, 3	amphichiral, noninvertible
	1, 4	amphichiral, noninvertible
	1, 2	chiral, invertible
	1, 2, 3, 4	and amphichiral, invertible

In the case of hyperbolic knots, the symmetry group must be finite and either cyclic or dihedral (Riley 1979, Kodama and Sakuma 1992, Hoste et al. 1998). The classification is slightly more complicated for nonhyperbolic knots. Furthermore, all prime knots with at most 8 crossings are either amphichiral or invertible (Hoste et al. 1998). Any symmetry of a prime alternating link must be visible up to flypes in any alternating diagram of the link (Bonahon and Siebermann, Menasco and Thistlethwaite 1993, Hoste et al. 1998).

The following tables (Hoste et al. 1998, Livingston and Moore 2026) give the numbers of -crossing hyperbolic prime knots belonging to cyclic symmetry groups (OEIS A052411 for and OEIS A052412 for ) and dihedral symmetry groups (OEIS A052415, OEIS A052416, OEIS A052417, OEIS A052418, OEIS A052420, and OEIS A052422). Of knots with 16 or fewer crossings, there are only one each having symmetry groups , , and (above left). There are only two knots with symmetry group , one hyperbolic (above right), and one a satellite knot. In addition, there are 2, 4, and 10 satellite knots having 14-, 15-, and 16-crossings, respectively, which belong to the dihedral group .

1	0	0	0	0
2	0	0	0	0
3	0	0	0	0
4	0	0	0	0
5	0	0	0	0
6	0	0	0	0
7	0	0	0	0
8	0	0	0	0
9	2	0	0	0
10	24	3	0	0
11	173	14	0	0
12	1047	57	0	0
13	6709	210	0	0
14	37177	712	0	2
15	224311	2268	1	0
16	1301492	7011	0	11

1	0	0	0	0	0	0	0	0	0	0	0	0
2	0	0	0	0	0	0	0	0	0	0	0	0
3	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	1	0	0	0	0	0	0	0	0
5	0	1	0	0	0	0	0	0	0	0	0	0
6	0	2	0	1	0	0	0	0	0	0	0	0
7	0	4	0	2	0	0	0	0	0	0	0	0
8	4	12	0	3	0	0	0	1	0	0	0	0
9	13	23	3	4	0	3	0	0	0	0	0	0
10	66	63	0	6	0	1	0	0	0	1	0	0
11	217	134	2	10	0	1	0	0	0	0	0	0
12	728	309	6	18	1	8	0	2	0	0	0	0
13	2391	647	1	21	1	3	0	2	0	0	0	0
14	7575	1463	4	31	2	2	0	0	0	0	1	0
15	23517	3065	49	53	3	12	0	2	0	4	0	0
16	73263	6791	15	89	0	10	1	8	1	1	0	1