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, Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... A trefoil knot. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... Circle illustration In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. ...

Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between inequivalent knots. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... A knot invariant is a useful tool in knot theory. ...

The abelianization of a knot group is always isomorphic to the infinite cyclic group Z. In mathematics, the derived group (or commutator subgroup) of a group G is the subgroup G1 generated by all the commutators of elements of G; that is, G1 = <[g,h] : g,h in G>. Note that the set of all commutators of the group is, generally, not a group (in... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

Examples

• 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 square knot and the granny knot have isomorphic knot groups, yet these two knots are inequivalent.