In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3,
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 .
Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant 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 that is isotopic to the identity and sends the first knot onto the second. Such a homeomorphism 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 of a knot group is always isomorphic to the infinite cyclic group Z; this follows because the abelianization agrees with the first homology group, which can be easily computed.
The knot group (or fundamental group of an oriented link in general) can be computed in the Wirtinger presentation by a relatively simple algorithm.
The unknot has knot group isomorphic to Z.
The trefoil knot has knot group isomorphic to the braid group B3. This group has the presentation
or
A (p,q)-torus knot has knot group with presentation
The figure eight knot has knot group with presentation
The square knot and the granny knot have isomorphic knot groups, yet these two knots are not equivalent.
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In mathematics, the braid group on n strands (denoted ), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see ). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see ); and in monodromy invariants of algebraic geometry.
vignette|Faire un nœud de trèfle (vidéo) vignette|Surface de Seifert associée à un nœud de trèfle : il en forme le bord. En théorie des nœuds, le nœud de trèfle est le nœud le plus simple après le nœud trivial. C'est le seul nœud premier à trois croisements. On peut aussi le décrire comme nœud torique de type (2,3), son mot dans le groupe de tresses étant σ13. Une autre description (liée à la précédente) est l'intersection de la sphère unité dans C2 avec la courbe plane complexe d'équation .
thumb|right|Représentation d’un nœud torique de type (3, 8). La théorie des nœuds est une branche de la topologie qui consiste en l'étude mathématique de courbes présentant des liaisons avec elles-mêmes, un « bout de ficelle » idéalisé en lacets. Elle est donc très proche de la théorie des tresses qui comporte plusieurs chemins ou « bouts de ficelle ». left|thumb|Nœuds triviaux La théorie des nœuds a commencé vers 1860 et avec des travaux de Carl Friedrich Gauss liés à l'électromagnétisme.
φ For all finite n ∈ N, there is a well-known isomorphism between the standard braid group Bn and the mapping class group π0Hn. This isomorphism has been exhaustively studied in literature, and generalized in many ways. For some basic topological reason, t ...
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