The braid group on n strands, denoted Bn and also known as the Artin braid group, is the group whose elements are equivalence classes of n-strand braids under ambient isotopy, with composition of braids as its group operation. Applications of braid groups include knot theory, where any knot can be represented as the closure of certain braids under a result known as Alexander's theorem; mathematical physics, where Artin's canonical presentation of the braid group corresponds to the Yang-Baxter equation; and monodromy invariants in algebraic geometry.
Facts
Classification
Object KindStructure or Algebraic Object 1 Connections
Attributed To
Source Wikipedia: Braid group
In Branch
Source Wikipedia: Braid group
Is Kind Of Object
Sources
1. Wikipedia: Braid group
History
Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974) they were already implicit in Adolf Hurwitz's work on monodromy from 1891.
History section
Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974) they were already implicit in Adolf Hurwitz's work on monodromy from 1891.
Attributed To: Emil Artin, Lead paragraph
), 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
In Branch: Knot Theory, Lead paragraph
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 § Basic properties); and in monodromy invariants of algebraic geometry.
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