In mathematics, the spherical braid group or Hurwitz braid group is a braid group on n strands. In comparison with the usual braid group, it has an additional group relation that comes from the strands being on the sphere. The group also has relations to the inverse Galois problem.
Definition The spherical braid group on n strands, denoted S B n {\displaystyle SB_{n}} or B n ( S 2 ) {\displaystyle B_{n}(S^{2})} , is defined as the fundamental group of the configuration space of the sphere:
B n ( S 2 ) = π 1 ( C o n f n ( S 2 ) ) . {\displaystyle B_{n}(S^{2})=\pi _{1}(\mathrm {Conf} _{n}(S^{2})).}
The spherical braid group has a presentation in terms of generators σ 1 , σ 2 , ⋯ , σ n − 1 {\displaystyle \sigma _{1},\sigma _{2},\cdots ,\sigma _{n-1}} with the following relations:
σ i σ j = σ j σ i {\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}} for | i − j | ≥ 2 {\displaystyle |i-j|\geq 2}
σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 {\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}} for 1 ≤ i ≤ n − 2 {\displaystyle 1\leq i\leq n-2} (the Yang–Baxter equation)
σ 1 σ 2 ⋯ σ n − 1 σ n − 1 σ n − 2 ⋯ σ 1 = 1 {\displaystyle \sigma _{1}\sigma _{2}\cdots \sigma _{n-1}\sigma _{n-1}\sigma _{n-2}\cdots \sigma _{1}=1}
The last relation distinguishes the group from the usual braid group.
References
