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The composition of the braids σ and τ is written as στ.. The set of all braids on four strands is denoted by .The above composition of braids is indeed a group operation. . The identity element is the braid consisting of four parallel horizontal strands, and the inverse of a braid consists of that braid which "undoes" whatever the first braid did, which is obtained by flipping a diagram ...
In the mathematical area of braid theory, the Dehornoy order is a left-invariant total order on the braid group, found by Patrick Dehornoy. [1] [2] Dehornoy's original discovery of the order on the braid group used huge cardinals, but there are now several more elementary constructions of it. [3]
This is a typical element of the braid group, which is used in the mathematical field of knot theory. In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs.
To be precise, the loop braid group on n loops is defined as the motion group of n disjoint circles embedded in a compact three-dimensional "box" diffeomorphic to the three-dimensional disk. A motion is a loop in the configuration space, which consists of all possible ways of embedding n circles into the 3-disk.
Loop braid group; M. Matsumoto's theorem (group theory) S. Spherical braid group
It follows from this definition and the fact that and are Eilenberg–MacLane spaces of type (,), that the unordered configuration space of the plane is a classifying space for the Artin braid group, and is a classifying space for the pure Artin braid group, when both are considered as discrete groups.
A group is a very general algebraic object and most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite non-abelian groups such as a braid group.
Brunnian braids form a subgroup of the braid group. Brunnian braids over the 2-sphere that are not Brunnian over the 2-disk give rise to non-trivial elements in the homotopy groups of the 2-sphere. For example, the "standard" braid corresponding to the Borromean rings gives rise to the Hopf fibration S 3 → S 2, and iteration of this (as in ...