Amphichiral Knot

An amphichiral knot is a knot that is capable of being continuously deformed into its own mirror image. More formally, a knot K is amphichiral (also called achiral or amphicheiral) if there exists an orientation-reversing homeomorphism of R^3 mapping K 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 4_1 (the figure eight knot), 6_3, 8_3, 8_9, 8_(12), 8_(17), 8_(18), 10_(17), 10_(33), 10_(37), 10_(43), 10_(45), 10_(79), 10_(81), 10_(88), 10_(99), 10_(109), 10_(115), 10_(118), and 10_(123) (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 (a) with n crossings, starting with n=3.

amph.A0524010, 1, 0, 1, 0, 5, 0, 13, 0, 58, 0, 274, 1, ...
+A0517670, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 0, 65, ...
-A0517680, 0, 0, 0, 0, 1, 0, 6, 0, 40, 0, 227, 1, ...
aA0524000, 1, 0, 1, 0, 4, 0, 7, 0, 17, 0, 41, 0, 113, ...

Prime amphichiral alternating knots can only exist for even n, 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 b_+ be the sum of positive exponents, and b_- the sum of negative exponents in the braid group B_n. If


then the knot corresponding to the closed braid b is not amphichiral (Jones 1985).

See also

Alternating Knot, Amphichiral, Braid Group, Chiral Knot, Invertible Knot, Knot Symmetry, Mirror Image

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Burde, G. and Zieschang, H. Knots, 2nd rev. ed. Berlin: de Gruyter, pp. 311-319, 2002.Haseman, M. G. "On Knots, with a Census of the Amphicheirals with Twelve Crossings." Trans. Roy. Soc. Edinburgh 52, 235-255, 1917.Haseman, M. G. "Amphicheiral Knots." Trans. Roy. Soc. Edinburgh 52, 597-602, 1918.Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1701936 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 A051767, A051768, A052400, and A052401 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Amphichiral Knot

Cite this as:

Weisstein, Eric W. "Amphichiral Knot." From MathWorld--A Wolfram Web Resource.

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