A symmetry of a knot
is a homeomorphism of
which maps
onto itself. More succinctly, a knot symmetry is a homeomorphism of the pair of spaces
. Hoste et al. (1998) consider
four types of symmetry based on whether the symmetry preserves or reverses orienting
of and ,

1. preserves ,
preserves
(identity operation),

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 knots with
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) give the numbers of -crossing knots belonging to cyclic symmetry groups (Sloane's A052411
for
and A052412 for ) and dihedral symmetry groups (Sloane's A052415
through 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 .

Bonahon, F. and Siebermann, L. "The Classification of Algebraic Links." Unpublished manuscript.Hoste, J.; Thistlethwaite,
M.; and Weeks, J. "The First Knots." Math. Intell.20, 33-48,
Fall 1998.Kodama K. and Sakuma, M. "Symmetry Groups of Prime Knots
Up to 10 Crossings." In Knot
90, Proceedings of the International Conference on Knot Theory and Related Topics,
Osaka, Japan, 1990 (Ed. A. Kawauchi.) Berlin: de Gruyter, pp. 323-340,
1992.Menasco, W. and Thistlethwaite, M. "The Classification of
Alternating Links." Ann. Math.138, 113-171, 1993.Riley,
R. "An Elliptic Path from Parabolic Representations to Hyperbolic Structures."
In Topology
of Low-Dimensional Manifolds, Proceedings, Sussex 1977 (Ed. R. Fenn).
New York: Springer-Verlag, pp. 99-133, 1979.Sloane, N. J. A.
Sequences A052411, A052412,
A052415, A052416,
A052417, A052418,
A052420, and A052422
in "The On-Line Encyclopedia of Integer Sequences."