An amphichiral knot is a knot that is capable of being continuously deformed into its own mirror image. More formally,
a knot 
is amphichiral (also called achiral or amphicheiral) if there exists an orientation-reversing
homeomorphism of 
mapping 
to itself (Hoste et al. 1998). (If the words "orientation-reversing"
are omitted, all knots are equivalent to their mirror
images.)
Knots on ten and fewer crossing can be tested in the Wolfram Language to see if they are amphichiral using the command KnotData[knot, "Amphichiral"].
There are 20 amphichiral knots having ten or fewer crossings, namely 
(the figure eight knot),
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
(Jones 1985), the first few of which are illustrated
above.
The following table gives the total number of prime amphichiral knots, number of 
amphichiral noninvertible prime knots,
amphichiral noninvertible prime knots,
and fully amphichiral invertible knots prime knots
(
)
with 
crossings, starting with 
.
| type | OEIS | counts |
| amph. | A052401 | 0, 1, 0, 1, 0, 5, 0, 13, 0, 58, 0, 274, 1, ... |
 | A051767 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 0, 65, ... |
 | A051768 | 0, 0, 0, 0, 0, 1, 0, 6, 0, 40, 0, 227, 1, ... |
 | A052400 | 0, 1, 0, 1, 0, 4, 0, 7, 0, 17, 0, 41, 0, 113, ... |
Prime amphichiral alternating knots can only exist for even 
,
but the 15-crossing nonalternating amphichiral knot illustrated above was discovered
by Hoste et al. (1998). It is the only known prime nonalternating amphichiral
knot with an odd number of crossings.
The HOMFLY polynomial is good at identifying amphichiral knots, but sometimes fails to identify knots which are not. No knot invariant which always definitively determines if a knot is amphichiral is known.
Let 
be the sum of positive exponents,
and 
the sum of negative exponents
in the braid group 
. If
 |
then the knot corresponding to the closed braid 
is not amphichiral (Jones 1985).
Knots." Math. Intell. 20, 33-48, Fall 1998.Jones,
V. "A Polynomial Invariant for Knots via von Neumann Algebras." Bull.
Amer. Math. Soc. 12, 103-111, 1985.Jones, V. "Hecke
Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126,
335-388, 1987.Sloane, N. J. A. Sequences