braid group

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  • Info In mathematics, the braid group on strands, denoted by, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group, Here, is a natural number; if, then is an infinite group. more...
  • This fact is also related to the braid groups well known in knot theory.
  • The second group can be thought of the same as with finite braid groups.
  • Since there are nevertheless several hard computational problems about braid groups, applications in cryptography have been suggested.
  • Her dissertation was titled Braid groups and their relationship to mapping class groups.
  • These are just the relations for the infinite braid group, together with the relations u = 0.
  • This sequence splits and therefore pure braid groups are realized as iterated semi-direct products of free groups.
  • Elements of the braid group can be represented more concretely by matrices.
  • Examples of such groups include braid groups and, more generally, Artin groups of finite Coxeter type.
  • A similar notion exists using a loop braid group.
  • Braid groups may also be given a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.
  • These algebras include certain quotients of the group algebras of braid groups.
  • So the term group-based cryptography refers mostly to cryptographic protocols that use infinite nonabelian groups such as a braid group.
  • Such groups may be described by explicit presentations, as was shown by, For an elementary treatment along these lines, see the article on braid groups.
  • As the number of cables increases, the number of crossing patterns increases, as described by the braid group.
  • The term braided comes from the fact that the braid group plays an important role in the theory of braided monoidal categories.
  • There is also a package called CHEVIE for GAP3 with special support for braid groups.
  • This result goes by the phrase "braid groups are linear."
  • Dehornoy's original discovery of the order on the braid group used huge cardinals, but there are now several more elementary constructions of it.
  • The basic operations which generate a loop braid group for n loops are exchanges of two adjacent loops, and passing one adjacent loop through another.
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