mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, especially in the area of

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

known as

knot theory In 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 joined so it cannot be und ...

, an invertible knot is a

knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...

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 i ...

. 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 fro ...

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 (mathematics), knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop (topology ...

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 sailing, rock climbing and caving as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under ...

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 discovered an infinite family of

pretzel knot A Pretzel knot may refer to: * Pretzel link: a concept in mathematics * Soft pretzel with garlic * Stafford knot: a rope knot used in sailing and heraldry {{Disambig ...

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

almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...

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: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

, 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

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

pretzel knot A Pretzel knot may refer to: * Pretzel link: a concept in mathematics * Soft pretzel with garlic * Stafford knot: a rope knot used in sailing and heraldry {{Disambig ...

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

pretzel knot A Pretzel knot may refer to: * Pretzel link: a concept in mathematics * Soft pretzel with garlic * Stafford knot: a rope knot used in sailing and heraldry {{Disambig ...

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

*Jablan, Slavik & Sazdanovic, Radmila
Basic graph theory: Non-invertible knot and links
, ''LinKnot''.

''Nrich.Maths.org''. {{Knot theory, state=collapsed

Background

Invertible knots

Strongly invertible knots

Non-invertible knots

See also

References

External links