by  S.Yu. Orevkov

Abstract:   P. Dehornoy [1], [2] constructed a right-invariant order on the braid group  Bn . The order is uniquely defined by the condition  $\beta_0\sigma_i\beta_1>1$  if  $\beta_0$  and  $\beta_1$  are words formed from  $\sigma_{i+1}^{±1}, \dots, \sigma_{n-1}^{±1}$.  A braid is said to be strongly positive if  $\alpha \beta \alpha^{-1}>1$  for all  $\alpha \in B_n$.  We prove that the braid  $\beta_0(\sigma_1 \sigma_2 ... \sigma_{n-1}) (\sigma_{n-1}\sigma_{n-2} ... \sigma_1)$  is strongly positive if the word  $\beta_0$ does not contain $\sigma_1^{±1}$.

We also give a geometric proof of the result of Burckel and Laver that the standard generators of the braid group are strongly positive. Finally, we discuss the relation between a right-invariant order and quasipositivity.

[D1]. P. Dehornoy  Trans. A.M.S. 345(1994), 115-150
[D2]. P. Dehornoy  Adv. Math. 125(1997), 200-235