Artin automorphism of a braid ($x_i\mapsto x_i{x}_{i+1}{x}_i^{-1},x_{i+1}\mapsto x_i$)

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 $⟨x_i\mid x_i=\varphi (x_i)⟩$.