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- Loop_braid_group abstract "The loop braid group is a mathematical group structure that is used in some models of theoretical physics to model the exchange of particles with loop-like topologies within three dimensions of space and time.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. The topology forces these generators to satisfy some relations, which determine the group.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. This becomes a group in the same way as loops in any space can be made into a group; first, we define equivalence classes of loops by letting paths g and h be equivalent iff they are related by a (smooth) homotopy, and then we define a group operation on the equivalence classes by concatenation of paths. Dahm was able to show that there is an injective homomorphism from this group into the automorphism group of the free group on n generators, so it is natural to identify the group with this subgroup of the automorphism group. One may also show that the loop braid group is isomorphic to the welded braid group, as is done for example in a paper by Baez, Wise, and Crans, which also gives some presentations of the loop braid group using the work of Lin.".
- Loop_braid_group wikiPageID "4543610".
- Loop_braid_group wikiPageLength "2538".
- Loop_braid_group wikiPageOutDegree "6".
- Loop_braid_group wikiPageRevisionID "621101930".
- Loop_braid_group wikiPageWikiLink Braid_group.
- Loop_braid_group wikiPageWikiLink Category:Braid_groups.
- Loop_braid_group wikiPageWikiLink Elementary_particle.
- Loop_braid_group wikiPageWikiLink Group_(mathematics).
- Loop_braid_group wikiPageWikiLink John_C._Baez.
- Loop_braid_group wikiPageWikiLink Theoretical_physics.
- Loop_braid_group wikiPageWikiLinkText "Loop braid group".
- Loop_braid_group wikiPageWikiLinkText "loop braid group".
- Loop_braid_group wikiPageUsesTemplate Template:Reflist.
- Loop_braid_group wikiPageUsesTemplate Template:Topology-stub.
- Loop_braid_group subject Category:Braid_groups.
- Loop_braid_group hypernym Structure.
- Loop_braid_group type Building.
- Loop_braid_group type Group.
- Loop_braid_group type Group.
- Loop_braid_group comment "The loop braid group is a mathematical group structure that is used in some models of theoretical physics to model the exchange of particles with loop-like topologies within three dimensions of space and time.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.".
- Loop_braid_group label "Loop braid group".
- Loop_braid_group sameAs Q17098636.
- Loop_braid_group sameAs m.0c81g1.
- Loop_braid_group sameAs Q17098636.
- Loop_braid_group wasDerivedFrom Loop_braid_group?oldid=621101930.
- Loop_braid_group isPrimaryTopicOf Loop_braid_group.