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

- 0₁ knot/Unknot - a simple un-knotted closed loop
- 3₁ knot/Trefoil knot - (2,3)-torus knot, the two loose ends of a common overhand knot joined together
- 4₁ knot/Figure-eight knot (mathematics) - a prime knot with a crossing number four
- 5₁ knot/Cinquefoil knot, (5,2)-torus knot, Solomon's seal knot, pentafoil knot - a prime knot with crossing number five which can be arranged as a {5/2} star polygon (pentagram)
- 5₂ knot/Three-twist knot - the twist knot with three-half twists
- 6₁ knot/Stevedore knot (mathematics) - a prime knot with crossing number six, it can also be described as a twist knot with four twists
- 6₂ knot - a prime knot with crossing number six
- 6₃ knot - a prime knot with crossing number six
- 7₁ knot, septafoil knot, (7,2)-torus knot - a prime knot with crossing number seven, which can be arranged as a {7/2} star polygon (heptagram)
- 7₄ knot, "endless knot"
- 8
_{18}knot, "carrick mat" - 10₁₆₁/Perko pair
- 12n242/(−2,3,7) pretzel knot
- (
*p*,*q*)-torus knot - a special kind of knot that lies on the surface of an unknotted torus in R^{3}

- Square knot (mathematics) - a composite knot obtained by taking the connected sum of a trefoil knot with its reflection
- Granny knot (mathematics) - a composite knot obtained by taking the connected sum of two identical trefoil knots

- 0
^{2}_{1}link/Unlink - equivalent under ambient isotopy to finitely many disjoint circles in the plane - 2
^{2}_{1}link/Hopf link - the simplest nontrivial link with more than one component; it consists of two circles linked together exactly once (L2a1) - 4
^{2}_{1}link/Solomon's knot (a two component "link" rather than a one component "knot") - a traditional decorative motif used since ancient times (L4a1) - 5
^{2}_{1}link/Whitehead link - two projections of the unknot: one circular loop and one figure eight-shaped loop intertwined such that they are inseparable and neither loses its form (L5a1) - Brunnian link - a nontrivial link that becomes trivial if any component is removed
- 6
^{3}_{2}link/Borromean rings - three topological circles which are linked and form a Brunnian link (L6a4) - L10a140 link - presumably the simplest non-Borromean Brunnian link
- Pretzel link - a Montesinos link with integer tangles

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**.

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, **R**^{3}. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of **R**^{3} upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

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.

In mathematics, a **knot** is an embedding of a circle *S*^{1} in 3-dimensional Euclidean space, **R**^{3}, 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 *S*^{ j} in *S*^{n}, 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.

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.

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.

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.

In knot theory, a **torus knot** is a special kind of knot that lies on the surface of an unknotted torus in **R**^{3}. 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.

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.

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.

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

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.

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.

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,.

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.

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.

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.

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