In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image (when identical while reversed). An oriented knot that is equivalent to its mirror image is an amphichiral knot, also called an achiral 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.
There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, reversible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible.^{[1]}
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.^{[2]} However, Tait's conjecture was proven true for prime, alternating knots.^{[3]}
Number of crossings | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | OEIS sequence |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Chiral knots | 1 | 0 | 2 | 2 | 7 | 16 | 49 | 152 | 552 | 2118 | 9988 | 46698 | 253292 | 1387166 | N/A |
Reversible knots | 1 | 0 | 2 | 2 | 7 | 16 | 47 | 125 | 365 | 1015 | 3069 | 8813 | 26712 | 78717 | A051769 |
Fully chiral knots | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 27 | 187 | 1103 | 6919 | 37885 | 226580 | 1308449 | A051766 |
Amphichiral knots | 0 | 1 | 0 | 1 | 0 | 5 | 0 | 13 | 0 | 58 | 0 | 274 | 1 | 1539 | A052401 |
Positive Amphichiral Noninvertible knots | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 6 | 0 | 65 | A051767 |
Negative Amphichiral Noninvertible knots | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 6 | 0 | 40 | 0 | 227 | 1 | 1361 | A051768 |
Fully Amphichiral knots | 0 | 1 | 0 | 1 | 0 | 4 | 0 | 7 | 0 | 17 | 0 | 41 | 0 | 113 | A052400 |
The simplest chiral knot is the trefoil knot, which was shown to be chiral by Max Dehn. All nontrivial torus knots are chiral. The Alexander polynomial cannot distinguish a knot from its mirror image, 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 there is no known polynomial knot invariant that can fully detect chirality.^{[4]}
A chiral knot that is invertible is classified as a reversible knot.^{[5]} Examples include the trefoil knot.
If a knot is not equivalent to its inverse or its mirror image, it is a fully chiral knot, for example the 9 32 knot.^{[5]}
An amphichiral knot is one which has an orientation-reversing self-homeomorphism of the 3-sphere, α, fixing the knot set-wise. All amphichiral alternating knots have even crossing number. The first amphichiral knot with odd crossing number is a 15-crossing knot discovered by Hoste et al.^{[3]}
If a knot is isotopic to both its reverse and its mirror image, it is fully amphichiral. The simplest knot with this property is the figure-eight knot.
If the self-homeomorphism, α, preserves the orientation of the knot, it is said to be positive amphichiral. This is equivalent to the knot being isotopic to its mirror. No knots with crossing number smaller than twelve are positive amphichiral and noninvertible .^{[5]}
If the self-homeomorphism, α, reverses the orientation of the knot, it is said to be negative amphichiral. This is equivalent to the knot being isotopic to the reverse of its mirror image. The noninvertible knot with this property that has the fewest crossings is the knot 8_{17}.^{[5]}