Knot group - Citizendia

In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a knot is an Embedding of a Circle in 3-dimensional Euclidean space, R 3 considered up to continuous deformations In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the The knot group of a knot K is defined as the fundamental group of the knot complement of K in R³,

$\pi_1(\mathbb{R}^3 \backslash K).$

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, 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 In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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 abelianization of a knot group is always isomorphic to the infinite cyclic group Z. In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

Examples

The unknot has knot group isomorphic to Z. The unknot arises in the mathematical theory of knots. Intuitively the unknot is a closed loop of rope without a Knot in it
The trefoil knot has knot group isomorphic to the braid group B₃. In Knot theory, the trefoil knot is the simplest nontrivial knot. In Mathematics, the braid group on n strands, denoted by B n, is a certain group which has an intuitive geometrical representation This group has the presentation $\langle x,y \mid x^2 = y^3 \rangle$ or $\langle a, b \mid aba = bab \rangle$ . In Mathematics, one method of defining a group is by a presentation.
A (p,q)-torus knot has knot group with presentation $\langle x,y \mid x^p = y^q \rangle$ . In Knot theory, a torus knot is a special kind of knot which lies on the surface of an unknotted Torus in R 3
The square knot and the granny knot have isomorphic knot groups, yet these two knots are inequivalent. The reef knot or square knot is a common and simple Binding knot. The granny knot is a Binding knot, used to secure a Rope or line around an object

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