Spherical braid group

The spherical braid group on n strands, denoted ${\displaystyle SB_{n}}$ or ${\displaystyle B_{n}(S^{2})}$, is defined as the fundamental group of the configuration space of the sphere:^[2][3] $${\displaystyle B_{n}(S^{2})=\pi _{1}(\mathrm {Conf} _{n}(S^{2})).}$$ The spherical braid group has a presentation in terms of generators ${\displaystyle \sigma _{1},\sigma _{2},\cdots ,\sigma _{n-1}}$ with the following relations:^[4]

• ${\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}$ for ${\displaystyle |i-j|\geq 2}$
• ${\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}}$ for ${\displaystyle 1\leq i\leq n-2}$ (the Yang–Baxter equation)
• ${\displaystyle \sigma _{1}\sigma _{2}\cdots \sigma _{n-1}\sigma _{n-1}\sigma _{n-2}\cdots \sigma _{1}=1}$

The last relation distinguishes the group from the usual braid group.