Amphichiral Knot

An amphichiral knot is a Knot which is capable of being continuously deformed into its own Mirror Image. The amphichiral knots having ten or fewer crossings are 04-001 (Figure-of-Eight Knot), 06-003, 08-003, 08-009, 08-012, 08-017, 08-018, 10-017,10-033, 10-037, 10-043, 10-045, 10-079, 10-081, 10-088, 10-099, 10-109, 10-115, 10-118, and 10-123 (Jones 1985). The HOMFLY Polynomial is good at identifying amphichiral knots, but sometimes fails to identify knots which are not. No complete invariant (an 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

$$b_+-3b_--n+1>0,$$

then the Knot corresponding to the closed Braid $b$ is not amphichiral (Jones 1985).

See also Amphichiral, Braid Group, Invertible Knot, Mirror Image

References

Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter, pp. 311-319, 1985.

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.