Invertible knot

In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant. An invertible link is the link equivalent of an invertible knot.

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

Number of invertible and non-invertible knots for each crossing number
Number of crossings	3	4	5	6	7	8	9	10	11	12	13	14	15	16	OEIS sequence
Non-invertible knots	0	0	0	0	0	1	2	33	187	1144	6919	38118	226581	1309875	A052402
Invertible knots	1	1	2	3	7	20	47	132	365	1032	3069	8854	26712	78830	A052403

It has long been known that most of the simple knots, such as the trefoil knot and the figure-eight knot are invertible. In 1962 Ralph Fox conjectured that some knots were non-invertible, but it was not proved that non-invertible knots exist until Hale Trotter discovered an infinite family of pretzel knots that were non-invertible in 1963.^[2] It is now known almost all knots are non-invertible.^[3]

Invertible knots

The simplest non-trivial invertible knot, the trefoil knot. Rotating the knot 180 degrees in 3-space about an axis in the plane of the diagram produces the same knot diagram, but with the arrow's direction reversed.

All knots with crossing number of 7 or less are known to be invertible. No general method is known that can distinguish if a given knot is invertible.^[4] The problem can be translated into algebraic terms,^[5] but unfortunately there is no known algorithm to solve this algebraic problem.

If a knot is invertible and amphichiral, it is fully amphichiral. The simplest knot with this property is the figure eight knot. A chiral knot that is invertible is classified as a reversible knot.^[6]

Strongly invertible knots

A more abstract way to define an invertible knot is to say there is an orientation-preserving homeomorphism of the 3-sphere which takes the knot to itself but reverses the orientation along the knot. By imposing the stronger condition that the homeomorphism also be an involution, i.e. have period 2 in the homeomorphism group of the 3-sphere, we arrive at the definition of a strongly invertible knot. All knots with tunnel number one, such as the trefoil knot and figure-eight knot, are strongly invertible.^[7]

Non-invertible knots

The non-invertible knot 8₁₇, the simplest of the non-invertible knots.

The simplest example of a non-invertible knot is the knot 8₁₇ (Alexander-Briggs notation) or .2.2 (Conway notation). The pretzel knot 7, 5, 3 is non-invertible, as are all pretzel knots of the form (2p + 1), (2q + 1), (2r + 1), where p, q, and r are distinct integers, which is the infinite family proven to be non-invertible by Trotter.^[2]

References

^ 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.
^ ^a ^b Trotter, H. F. (1963), "Non-invertible knots exist", Topology, 2 (4): 275–280, doi:10.1016/0040-9383(63)90011-9, MR 0158395.
^ Murasugi, Kunio (2007), Knot Theory and Its Applications, Springer, p. 45, ISBN 9780817647186.
^ Weisstein, Eric W. "Invertible Knot". MathWorld. Accessed: May 5, 2013.
^ Kuperberg, Greg (1996), "Detecting knot invertibility", Journal of Knot Theory and Its Ramifications, 5 (2): 173–181, arXiv:q-alg/9712048, doi:10.1142/S021821659600014X, MR 1395778, S2CID 15295630.
^ Clark, W. Edwin; Elhamdadi, Mohamed; Saito, Masahico; Yeatman, Timothy (2013), "Quandle colorings of knots and applications", Journal of Knot Theory and Its Ramifications, 23 (6), arXiv:1312.3307, Bibcode:2013arXiv1312.3307C, doi:10.1142/S0218216514500357, PMC 4610146, PMID 26491208.
^ Morimoto, Kanji (1995), "There are knots whose tunnel numbers go down under connected sum", Proceedings of the American Mathematical Society, 123 (11): 3527–3532, doi:10.1090/S0002-9939-1995-1317043-4, JSTOR 2161103, MR 1317043. See in particular Lemma 5.

External links

Jablan, Slavik & Sazdanovic, Radmila. Basic graph theory: Non-invertible knot and links Archived 2011-01-18 at the Wayback Machine, LinKnot.
Explanation with a video, Nrich.Maths.org.

[htw-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.

[trotter-2] Trotter, H. F. (1963), "Non-invertible knots exist", Topology, 2 (4): 275–280, doi:10.1016/0040-9383(63)90011-9, MR 0158395.

[3] Murasugi, Kunio (2007), Knot Theory and Its Applications, Springer, p. 45, ISBN 9780817647186.

[mathworld-4] Weisstein, Eric W. "Invertible Knot". MathWorld. Accessed: May 5, 2013.

[5] Kuperberg, Greg (1996), "Detecting knot invertibility", Journal of Knot Theory and Its Ramifications, 5 (2): 173–181, arXiv:q-alg/9712048, doi:10.1142/S021821659600014X, MR 1395778, S2CID 15295630.

[6] Clark, W. Edwin; Elhamdadi, Mohamed; Saito, Masahico; Yeatman, Timothy (2013), "Quandle colorings of knots and applications", Journal of Knot Theory and Its Ramifications, 23 (6), arXiv:1312.3307, Bibcode:2013arXiv1312.3307C, doi:10.1142/S0218216514500357, PMC 4610146, PMID 26491208.

[7] Morimoto, Kanji (1995), "There are knots whose tunnel numbers go down under connected sum", Proceedings of the American Mathematical Society, 123 (11): 3527–3532, doi:10.1090/S0002-9939-1995-1317043-4, JSTOR 2161103, MR 1317043. See in particular Lemma 5.

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