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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 embeddingof $B_{m+1}$ into the symplectic mapping class group of $Q$ due to Khovanov andSeidel, and a natural lifting homomorphism. In order to show that the composedhomomorphism is still injective, we use a partially linearized variant of theChekanov--Eliashberg dga for Legendrians which lie above one another in $Q\times \mathbb{R}$, reducing the proof to Floer homology. As corollaries weobtain a contribution to the contact isotopy problem for $Q \times S^1$, aswell as the fact that in dimension $4$, the lifting homomorphism embeds thesymplectic mapping class group of $Q$ into the contact mapping class group of$Q \times S^1$.

On the contact mapping class group of the contactization of the $A_m$- Milnor fiber