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]}

## Contents

## Background

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

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 simplest example of a non-invertible knot is the knot 8_{17} (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 (2*p* + 1), (2*q* + 1), (2*r* + 1), where *p*, *q*, and *r* are distinct integers, which is the infinite family proven to be non-invertible by Trotter.^{[2]}

## See also

## 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, archived from the original (PDF) on 2013-12-15.- ^
^{a}^{b}Trotter, H. F. (1963), "Non-invertible knots exist",*Topology*,**2**: 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.**^**Clark, W. Edwin; Elhamdadi, Mohamed; Saito, Masahico; Yeatman, Timothy (2013),*Quandle colorings of knots and applications*, arXiv:1312.3307 , Bibcode:2013arXiv1312.3307C.**^**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,
*LinKnot*. - Explanation with a video,
*Nrich.Maths.org*.