In mathematics, 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 ' ...

is an

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is g ...

of a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

into 3-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R³, :

\pi_1(\mathbb^3 \setminus K).

Other conventions consider knots to be embedded in the 3-sphere, in which case the knot group is the fundamental group of its complement in

S^3

Properties

Two equivalent knots have isomorphic knot groups, so the knot group 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 ...

and can be used to distinguish between certain pairs of inequivalent knots. This is because an equivalence between two knots is a self-homeomorphism of

\mathbb^3

that is isotopic to the identity and sends the first knot onto the second. Such a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

restricts onto a homeomorphism of the complements of the knots, and this restricted homeomorphism induces an isomorphism of fundamental groups. However, it is possible for two inequivalent knots to have isomorphic knot groups (see below for an example). The

abelianization In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...

of a knot group is always isomorphic to the infinite

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

Z; this follows because the abelianization agrees with the first

homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...

, which can be easily computed. The knot group (or fundamental group of an oriented link in general) can be computed in the

Wirtinger presentation In mathematics, especially in group theory, a Wirtinger presentation is a finite presentation where the relations are of the form wg_iw^ = g_j where w is a word in the generators, \. Wilhelm Wirtinger observed that the complements of knots in 3-sp ...

by a relatively simple algorithm.

Examples

*The

unknot In the mathematical theory of knots, the unknot, not knot, 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, unknotted. To a knot theorist, an unknot is any embe ...

has knot group isomorphic to Z. *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 ...

has knot group isomorphic to the

braid group A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...

''B''₃. This group has the

presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...

\langle x,y \mid x^2 = y^3 \rangle

\langle a, b \mid aba = bab \rangle.

*A (''p'',''q'')-

torus knot 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 cop ...

has knot group with presentation ::

\langle x,y \mid x^p = y^q \rangle.

*The figure eight knot has knot group with presentation ::

\langle x,y \mid yxy^xy=xyx^yx\rangle

* The square knot and the

granny knot The granny knot is a binding knot, used to secure a rope or line around an object. It is considered inferior to the reef knot (square knot), which it superficially resembles. Neither of these knots should be used as a bend knot for attaching tw ...

have isomorphic knot groups, yet these two knots are not equivalent.

Properties

Examples

See also

Further reading