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.

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 H. F. Trotter discovered an infinite family of pretzel knots that were non-invertible in 1963. It is now known almost all knots are non-invertible.

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. The problem can be translated into algebraic terms, 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.

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.

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.

See also

• Chiral knot

External links

• Jablan, Slavik & Sazdanovic, Radmila. Basic graph theory: Non-invertible knot and links, LinKnot.
• Explanation with a video, Nrich.Maths.org.