mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the spherical braid group or Hurwitz braid group is a

braid group In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...

on strands. In comparison with the usual braid group, it has an additional group relation that comes from the strands being on the

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

. The group also has relations to the

inverse Galois problem In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There ...

Definition

The spherical braid group on strands, denoted

SB_n

B_n(S^2)

, is defined as the

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

of the configuration space of the sphere:

B_n(S^2) = \pi_1(\mathrm_n(S^2)).

The spherical braid group has a

presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...

in terms of generators

\sigma_1, \sigma_2, \cdots, \sigma_

with the following relations: *

\sigma_i \sigma_j = \sigma_j \sigma_i

for

, i-j,  \geq 2

\sigma_i \sigma_ \sigma_i = \sigma_ \sigma_i \sigma_

for

1 \leq i \leq n - 2

(the

Yang–Baxter equation In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their ...

) *

\sigma_1 \sigma_2 \cdots \sigma_ \sigma_ \sigma_ \cdots \sigma_ = 1

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

References

Braid groups {{Math-stub