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). TheHOMFLY 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 
be the Sum of Positive exponents, and 
the Sum of Negative exponents in the BraidGroup 
. If
then the Knot corresponding to the closed Braid
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.
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