## braid groups

30 examples (0.03 sec)
• 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.
• Braid groups find applications in knot theory, since any knot may be represented as the closure of certain braids.
• 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.