Palindromic Braids

The braid group $B_{n}$, endowed with Artin's presentation, admits an antiautomorphism $B_{n} \to B_{n}$, such that $v \mapsto \bar{v}$ is defined by reading braids in reverse order (from right to left instead of left to right). We prove that the map $B_{n} \to B_{n}$, $v \mapsto v \bar{v}$ is injective. We also give some consequences arising due to this injectivity.