Definition:Braid Group
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Description
The braid group is a group that has a intuitive geometric interpretation as a number of strands, where the group operation on these strands is to intertwine them.
Generators
The generators of the braid group are elements $\sigma_i$, which intertwine strands $i$ and $i+1$ in such a way that strand $i$ runs above strand $i+1$.
Definition
The braid group on $n$ strands is generated by $\sigma_1, \sigma_2, \ldots, \sigma_{n-1}$ and the following relations:
- $\sigma_i \sigma_j = \sigma_j\sigma_i, \forall i, j: |i-j| \ge 2$
- $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \forall i \in \{ 1, 2, \ldots, n-2 \}$
Examples
Generator $\sigma_i$ and the inverse generator $\sigma_i^{-1}$ acting on strands $s_i, s_{i+1}$:
Relation 1. can be pictured like this:
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Relation 2. can be pictured like this:
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