List of mathematical knots and links

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A table of all prime knots with seven crossings or fewer (not including mirror images). Knot table.svg
A table of all prime knots with seven crossings or fewer (not including mirror images).

This article contains a list of mathematical knots and links . See also list of knots, list of geometric topology topics.



Prime knots


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Unknot loop seen as a trivial knot

In the mathematical theory of knots, the unknot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic to a geometrically round circle, the standard unknot.

Knot theory study of mathematical knots

In topology, knot theory is the study of 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 together so that it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Prime knot non-trivial knot which cannot be written as the knot sum of two non-trivial knots

In knot theory, a prime knot or prime link is a knot that is, in a certain sense, undecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite knots or composite links. It can be a nontrivial problem to determine whether a given knot is prime or not.

Knot (mathematics) embedding of a circle in R³

In mathematics, a knot is an embedding of a circle S1 in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies). A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of Sj in Sn, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory, and has many simple relations to graph theory.

Trefoil knot Simplest non-trivial closed knot with three crossings

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 knot, the trefoil is fundamental to the study of mathematical knot theory.

Borromean rings three rings or loops, no two of which are linked together, yet all three are interlinked

In mathematics, the Borromean rings consist of three topological circles which are linked but where removing any one ring leaves the other two unconnected. In other words, no two of the three rings are linked with each other as a Hopf link, but nonetheless all three are linked. The Borromean rings are one of a class of such links called Brunnian links.

Link (knot theory) concept in mathematical knot theory

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link.

Torus knot knot which lies on the surface of a torus in 3-dimensional space

In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q are not coprime. A torus knot is trivial if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.

Hopf link prime link; simplest nontrivial link

In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf.

Brunnian link interlinked multi-loop construction where cutting one loop frees all the others; simplest is Borromean Rings

In knot theory, a branch of topology, a Brunnian link is a nontrivial link that becomes a set of trivial unlinked circles if any one component is removed. In other words, cutting any loop frees all the other loops.

Unlink link that consists of finitely many unlinked unknots

In the mathematical field of knot theory, the unlink is a link that is equivalent to finitely many disjoint circles in the plane.

Tricolorability knot theory problem

In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non-isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial.

Satellite knot

In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots and Whitehead doubles. A satellite link is one that orbits a companion knot K in the sense that it lies inside a regular neighborhood of the companion.

Crossing number (knot theory) integer-valued knot invariant; least number of crossings in a knot diagram

In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot. It is a knot invariant.

In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Bachelor's thesis,.

Twist knot

In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. The twist knots are an infinite family of knots, and are considered the simplest type of knots after the torus knots.

Unknotting number

In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself to untie it. If a knot has unknotting number , then there exists a diagram of the knot which can be changed to unknot by switching crossings. The unknotting number of a knot is always less than half of its crossing number.

L10a140 link minimal Brunnian link which is not equivalent to the Borromean rings

In the mathematical theory of knots, L10a140 is the name in the Thistlethwaite link table of a link of three loops, which has ten crossings between the loops when presented in its simplest visual form. It is of interest because it is presumably the simplest link which possesses the Brunnian property — a link of connected components that, when one component is removed, becomes entirely unconnected — other than the six-crossing Borromean rings.