Chiral knot
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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.
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[edit] 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]
| 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 knots | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 6 | 0 | 65 | A051767 |
| Negative Amphichiral 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 |
[edit] Chiral knot
 The left handed trefoil knot. |
 The right handed trefoil knot. |
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 Vk(q) ≠ Vk(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]
[edit] Reversible knot
A chiral knot that is invertible is classified as a reversible knot.[4]
[edit] Fully chiral knot
If a knot is not equivalent to its inverse or its mirror image, it is a fully chiral knot. [4]
[edit] Amphichiral knot
An amphichiral knot is one which has an orientation-reversing homeomorphism of the 3-sphere to itself fixing the knot. All amphichiral alternating knots have even crossing number. The only amphichiral knot with odd crossing number is a 15-crossing knot discovered by Hoste et al. [2]
[edit] 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.
[edit] 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]
[edit] Negative amphichiral
If a knot is different from its inverse and its mirror, but equivalent to the inverse of its mirror, then it is a negative amphichiral knot. The first knot with this property is the knot 817. [4]
[edit] References
- ^ History of Knot Theory and Certain Applications of Knots and Links
- ^ a b Weisstein, Eric W. "Amphichiral Knot." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AmphichiralKnot.html
- ^ http://arxiv.org/pdf/hep-th/9401095.pdf
- ^ a b c d Three Dimensional Invariants - Knot Atlas
