Braids: A Survey

By J. Birman, Tara E. Brendle. 2004

182 citations recorded by the source corpus.

Publisher Summary Crossings are suggested as they are in a picture of a highway overpass on a map. The identity braid has a canonical representation in which two strands never cross. Multiplication of braids is by juxtaposition, concatenation, isotopy, and rescaling. This chapter discusses Artin's braid group, B n and its role in knot theory. The chapter illustrates ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. Artin's braid group is naturally isomorphic to the mapping class group of an n-times punctured disc. The chapter also illustrates the topological concept of a braid and of a group of braids via the notion of a configuration space. It then outlines the new developments in the area of B n mapping, thus illustrating how to pass from diffeomorphisms to geometric braids and back again..

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e29a88b46c3e8d874cef0e419d39cfa9012e2b23

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Injections of Artin groups (2005, unknown): “n) is better known as the braid group on n + 1 strands. Artin’s classical work on braids identiﬁes A(An)/Z with the the subgroup of Mod(S) consisting of elements which ﬁx one particular puncture (see [3]). The pure braid group on n+1 strands P(An) is deﬁned as the kernel of the map from A(An) to the”
A new twist on Lorenz links (2007, background): “For example, see [2] for a proof that this mapping class group is isomorphic to Artin’s braid group Bt .”
GEOMETRIC PRESENTATIONS FOR THE PURE BRAID GROUP (2006, methodology): “SB = 1 when |B| = 1 SBSC = SCSB when B and C are compatible SBCDSBSCSD = SCBSBDSDC when (B,C,D) is admissible Notice that if we choose isotopy classes of convex curves a, b, c, d, x, y and z so that they surround the puncture sets BCD, B, C, D, BC, BD, and CD, respectively, then these seven curves c”