Braid group ($B_n$)

The group of braids on $n$ strands, composed by stacking, with adjacent-crossing generators ${\sigma }_i$ and the braid group relations. It retains over/under crossings and is not a permutation group.

A positive elementary braid acts on the free group of punctured-disk meridians by the displayed substitution, fixing other generators. The inverse sends $x_i$ to $x_{i+1}$ and $x_{i+1}$ to $x_{i+1}^{-1}{x}_i{x}_{i+1}$. Applying Seifert-van Kampen theorem at the crossings and closing the braid gives the knot-complement presentation $⟨x_i\mid x_i=\varphi (x_i)⟩$.

The generators ${\sigma }_i$ commute when $|i-j|>1$ and satisfy ${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$. Unlike a symmetric-group presentation, there is no ${\sigma }_i^2=1$ relation.