Artin automorphism of a braid
= Artin automorphism of a braid
{c}
{title2=$x_i\mapsto x_ix_{i+1}x_i^{-1},\quad x_{i+1}\mapsto x_i$}
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 $x_{i+1}^{-1}x_ix_{i+1}$. Applying <Seifert-van Kampen theorem> at the crossings and closing the braid gives the knot-complement presentation $\langle x_i\mid x_i=\phi(x_i)\rangle$.