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.
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.
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. However, Tait's conjecture was proven true for prime, alternating knots.
|Number of crossings||3||4||5||6||7||8||9||10||11||12||13||14||15||16||OEIS sequence|
|Fully chiral knots||0||0||0||0||0||0||2||27||187||1103||6919||37885||226580||1308449||A051766|
|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|
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 there is no known polynomial knot invariant which can fully detect chirality.
Fully chiral knot
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.
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.
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 knot with this property that has the fewest crossings is the knot 817.
- Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998), "The first 1,701,936 knots" (PDF), The Mathematical Intelligencer 20 (4): 33–48, doi:10.1007/BF03025227, MR 1646740.
- Jablan, Slavik & Sazdanovic, Radmila. "History of Knot Theory and Certain Applications of Knots and Links", LinKnot.
- Weisstein, Eric W., "Amphichiral Knot", MathWorld. Accessed: May 5, 2013.
- "Chirality of Knots 942 and 1071 and Chern-Simons Theory" by P. Ramadevi, T. R. Govindarajan, and R. K. Kaul
- "Three Dimensional Invariants", The Knot Atlas.