  We construct an embedding of the full braid group on $m+1$ strands $B_{m+1}$,
$m \geq 1$, into the contact mapping class group of the contactization $Q
\times S^1$ of the $A_m$-Milnor fiber $Q$. The construction uses the embedding
of $B_{m+1}$ into the symplectic mapping class group of $Q$ due to Khovanov and
Seidel, and a natural lifting homomorphism. In order to show that the composed
homomorphism is still injective, we use a partially linearized variant of the
Chekanov--Eliashberg dga for Legendrians which lie above one another in $Q
\times \mathbb{R}$, reducing the proof to Floer homology. As corollaries we
obtain a contribution to the contact isotopy problem for $Q \times S^1$, as
well as the fact that in dimension $4$, the lifting homomorphism embeds the
symplectic mapping class group of $Q$ into the contact mapping class group of
$Q \times S^1$.
