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The braid group on n {\displaystyle n} letters, denoted B n {\displaystyle B_{n}} , is defined as follows:
⟨ s 1 , s 2 , … , s n − 1 ∣ s i s i + 1 s i = s i + 1 s i s i + 1 ∀ 1 ≤ i ≤ n − 2 , s i s j = s j s i ∀ | i − j | > 1 ⟩ {\displaystyle \langle s_{1},s_{2},\dots ,s_{n-1}\mid s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}\ \forall \ 1\leq i\leq n-2,s_{i}s_{j}=s_{j}s_{i}\ \forall \ |i-j|>1\rangle } .
letters, denoted 
, is defined as follows:
.