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

Abstract

Dehornoy constructed a right invariant order on the braid group B _n uniquely defined by the condition $\beta _0 \sigma _i \beta _1 >1{\text{ if }}\beta _0 ,\beta _1$ are words in $\sigma _{i + 1}^{ \pm 1} ,...,\sigma _{n - 1}^{ \pm 1}$. 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 $\beta _0 \left( {\sigma _1 \sigma _2 ...\sigma _{n - 1} } \right)\left( {\sigma _{n - 1} \sigma _{n - 2} ...\sigma _1 } \right)$ 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.