Chiral knot
In the
There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, invertible, positively amphicheiral noninvertible, negatively amphicheiral noninvertible, and fully amphicheiral invertible.[1]
Background
The possible chirality of certain knots was suspected since 1847 when
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 |
Invertible 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 |
Amphicheiral knots | 0 | 1 | 0 | 1 | 0 | 5 | 0 | 13 | 0 | 58 | 0 | 274 | 1 | 1539 | A052401 |
Positive Amphicheiral Noninvertible knots | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 6 | 0 | 65 | A051767 |
Negative Amphicheiral Noninvertible knots | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 6 | 0 | 40 | 0 | 227 | 1 | 1361 | A051768 |
Fully Amphicheiral knots | 0 | 1 | 0 | 1 | 0 | 4 | 0 | 7 | 0 | 17 | 0 | 41 | 0 | 113 | A052400 |
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The left-handed trefoil knot.
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The right-handed trefoil knot.
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 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 there is no known polynomial knot invariant that can fully detect chirality.[7]
Invertible knot
A chiral knot that can be smoothly deformed to itself with the opposite orientation is classified as a invertible knot.[8] Examples include the trefoil knot.
Fully chiral 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.[8]
Amphicheiral knot
An amphicheiral knot is one which has an orientation-reversing self-homeomorphism of the 3-sphere, α, fixing the knot set-wise. All amphicheiral alternating knots have even crossing number. The first amphicheiral knot with odd crossing number is a 15-crossing knot discovered by Hoste et al.[6]
Fully amphicheiral
If a knot is isotopic to both its reverse and its mirror image, it is fully amphicheiral. The simplest knot with this property is the figure-eight knot.
Positive amphicheiral
If the self-homeomorphism, α, preserves the orientation of the knot, it is said to be positive amphicheiral. This is equivalent to the knot being isotopic to its mirror. No knots with crossing number smaller than twelve are positive amphicheiral and noninvertible .[8]
Negative amphicheiral
If the self-homeomorphism, α, reverses the orientation of the knot, it is said to be negative amphicheiral. 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 817.[8]
References
- S2CID 18027155, archived from the original(PDF) on 2013-12-15.
- .
- S2CID 123957148.
- S2CID 124014620.
- S2CID 18027155.
- ^ a b Weisstein, Eric W. "Amphichiral Knot". MathWorld. Accessed: May 5, 2013.
- S2CID 119143024.
- ^ a b c d "Three Dimensional Invariants", The Knot Atlas.