In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image. An oriented knot that is equivalent to its mirror image is an amphichiral knot, also called an achiral knot or amphicheiral knot. The chirality of a knot is a knot invariant. A knot's chirality can be further classified depending on whether or not it is invertible.

Background

The chirality of certain knots was long suspected, and was proven by Max Dehn in 1914. P. G. Tait conjectured that all amphichiral knots had even crossing number, but a counterexample was found by Morwen Thistlethwaite et al. in 1998. ^[1] However, Tait's conjecture was proven true for prime, alternating knots. ^[2]

Chiral knot

<gallery caption="Both possible trefoil knots." widths="80px" heights="80px" align="right" perrow="3"> Image:TrefoilKnot-02.png|The left handed trefoil knot. Image:TrefoilKnot_01.svg|The right handed trefoil knot. </gallery> The simplest chiral knot is the trefoil knot, which was shown to be chiral by Max Dehn. All torus knots are chiral. The Alexander polynomial cannot detect the chirality of a knot, but the Jones polynomial can in some cases; if V_k(q) ? V_k(q^-1), then the knot is chiral, however the converse is not true. The HOMFLY polynomial is even better at detecting chirality, but no known knot invariant is known which can fully detect chirality. ^[3]

Reversible knot

A chiral knot that is invertible is classified as a reversible knot.^[4]

Fully chiral knot

If a knot is not equivalent to its inverse or its mirror image, it is a fully chiral knot. ^[4]

Amphichiral knot

The figure eight knot is the simplest amphichiral knot.

Fully amphichiral

If a knot is equivalent to both its inverse and its mirror, it is fully amphichiral. The first knot with this property is the figure eight knot.

Positive amphichiral

A positive amphichiral knot is one that is different from its inverse but equivalent to its mirror. No knots with crossing number ? 11 are positive amphichiral. ^[4]

Negative amphichiral

The first negative amphichiral knot.

References

See also

• Invertible knot
• Knot invariant
• Knot theory

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