Chiral knot

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 amphicheiral 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, invertible, positively amphicheiral noninvertible, negatively amphicheiral noninvertible, and fully amphicheiral invertible.^[1]

Background

The possible chirality of certain knots was suspected since 1847 when Johann Listing asserted that the trefoil was chiral,^[2] and this was proven by Max Dehn in 1914. P. G. Tait found all amphicheiral knots up to 10 crossings and conjectured that all amphicheiral knots had even crossing number. Mary Gertrude Haseman found all 12-crossing and many 14-crossing amphicheiral knots in the late 1910s.^[3][4] But a counterexample to Tait's conjecture, a 15-crossing amphicheiral knot, was found by Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks in 1998.^[5] However, Tait's conjecture was proven true for prime, alternating knots.^[6]

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
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

• 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 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.^[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

The figure-eight knot is the simplest 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

The first negative amphicheiral knot.

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

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, S2CID 18027155, archived from the original (PDF) on 2013-12-15.
2. Przytycki, Józef H. (1998). "Classical Roots of Knot Theory". Chaos, Solitons and Fractals. 9 (4/5): 531–45. Bibcode:1998CSF.....9..531P. doi:10.1016/S0960-0779(97)00107-0.
3. Haseman, Mary Gertrude (1918). "XI.—On Knots, with a Census of the Amphicheirals with Twelve Crossings". Trans. R. Soc. Edinb. 52 (1): 235–55. doi:10.1017/S0080456800012102. S2CID 123957148.
4. Haseman, Mary Gertrude (1920). "XXIII.—Amphicheiral Knots". Trans. R. Soc. Edinb. 52 (3): 597–602. doi:10.1017/S0080456800004476. S2CID 124014620.
5. Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998). "The First 1,701,936 Knots". Math. Intell. 20 (4): 33–48. doi:10.1007/BF03025227. S2CID 18027155.
6. Weisstein, Eric W. "Amphichiral Knot". MathWorld. Accessed: May 5, 2013.
7. Ramadevi, P.; Govindarajan, T.R.; Kaul, R.K. (1994). "Chirality of Knots 9₄₂ and 10₇₁ and Chern-Simons Theory"". Mod. Phys. Lett. A. 9 (34): 3205–18. arXiv:hep-th/9401095. Bibcode:1994MPLA....9.3205R. doi:10.1142/S0217732394003026. S2CID 119143024.
8. "Three Dimensional Invariants", The Knot Atlas.