Chiral knot

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.^[1]

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.^[2] However, Tait's conjecture was proven true for prime, alternating knots.^[3]

Number of knots of each type of chirality for each crossing number

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

• Both possible trefoil knots.
• 
  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 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 which can fully detect chirality.^[4]

Reversible knot

A chiral knot that is invertible is classified as a reversible knot.^[5] 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.^[5]

Amphichiral knot

The figure-eight knot is the simplest amphichiral 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.^[3]

Fully amphichiral

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.

Positive amphichiral

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.^[5]

Negative amphichiral

The first negative amphichiral knot.

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.^[5]

References

1. 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.
2. Jablan, Slavik & Sazdanovic, Radmila. "History of Knot Theory and Certain Applications of Knots and Links", LinKnot.
3. Weisstein, Eric W. "Amphichiral Knot". MathWorld. Accessed: May 5, 2013.
4. "Chirality of Knots 9₄₂ and 10₇₁ and Chern-Simons Theory" by P. Ramadevi, T. R. Govindarajan, and R. K. Kaul
5. "Three Dimensional Invariants", The Knot Atlas.