S.Yu. Orevkov

A braid $b$ is called quasipositive if $b=\prod_{j=1}^k a_j \sigma_1 a_j^{-1}$ for some braids $a_j$ (here $\sigma_1$ is a standard generator of the braid group. In a series of previouse papers we exploited the observation that the quasipositivity of a certain braid provides a necessary condition for the reailizability of a given isotopy type by a plane real algebraic curve of a given degree. As a test for the quasipositivity we used Murasugi-Tristram signature inequality, elementary argumets based on linking numbers, or Garside normal form for braid with three strings.

Here we propose a new simple test for the quasipositivity and give an example when it gives some new restrictions for real algebraic curves of 7th degree. The test is based on the following elementary observation. Suppose we are interested if a given braid $b$ is quasipositive. If it is then the number $k$ of the factors in any quasipositive presentation is just the image of $b$ under the abelianization $B_m\to Z$. Let $\rho:B_m\to SU(n)$ be any unitary representation. Then the matrix $\rho(b)$ is a product of $k$ matrices each of which is conjugated to $\rho(\sigma_1)$. A neseccary and sufficient condition for a given matrix to be presented as a product of matrices from given conjugacy classes was obtained by Agnihotri and Woodward.