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# Amphichiral Knot

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).

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

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## References

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 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."

Amphichiral Knot

## Cite this as:

Weisstein, Eric W. "Amphichiral Knot." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AmphichiralKnot.html