Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity

Dehornoy constructed a right invariant order on the braid group $B_n$ uniquely defined by the condition $\beta_0\sigma_i\beta_1>1$, if $\beta_0,\beta_1$ are words in $\sigma_{i+1}^{\pm 1},\dots,\sigma_{n-1}^{\pm 1}$. A braid is called strongly positive if $\alpha\beta\alpha^{-1}>1$ for any $\alpha\in B_n$. In the present paper it is proved that the braid $\beta_0(\sigma_1\sigma_2\dots\sigma_{n-1})(\sigma_{n-1}\sigma_{n-2}\dots\sigma_1)$ is strongly positive if the word $\beta_0$ does not contain $\sigma_1^{\pm 1}$. We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity.