mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, especially in the area of

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

known as

knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...

, an invertible knot is a

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

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 In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...

. An invertible link is the link equivalent of an invertible knot. There are only five knot symmetry types, indicated by

chirality Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...

and invertibility: fully chiral, reversible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible..

Background

It has long been known that most of the simple knots, such as the

trefoil knot In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest ...

and the

figure-eight knot The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...

are invertible. In 1962

Ralph Fox Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the ''Golden Age of differential topology'', and he played ...

conjectured that some knots were non-invertible, but it was not proved that non-invertible knots exist until

Hale Trotter Hale Freeman Trotter (30 May 1931 – 17 January 2022)biographical information from ''American Men and Women of Science'', Thomson Gale 2004 was a Canadian-American mathematician, known for the Lie–Trotter product formula, the Steinhaus–Johns ...

discovered an infinite family of

pretzel knot A Pretzel knot may refer to: * Pretzel link: a concept in mathematics * Soft pretzel A pretzel (), from German pronunciation, standard german: Breze(l) ( and French / Alsatian: ''Bretzel'') is a type of baked bread made from dough that is c ...

s that were non-invertible in 1963.. It is now known

almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...

knots are non-invertible.

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. Accessed: May 5, 2013. 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 Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...

, 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 In mathematics, the tunnel number of a knot, as first defined by Bradd Clark, is a knot invariant, given by the minimal number of arcs (called tunnels) that must be added to the knot so that the complement becomes a handlebody. The tunnel number c ...

one, such as the

and

, are strongly invertible.. See in particular Lemma 5.

Non-invertible knots

The simplest example of a non-invertible knot is the knot 8₁₇ (Alexander-Briggs notation) or .2.2 ( Conway notation). The

7, 5, 3 is non-invertible, as are all

s 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.

References

External links