Quasipositivity problem for 3-braids

by  S.Yu. Orevkov

Abstract:

Let  G  be a group and  X  a fixed set of its elements. An element  g  of  G  is called   X-quasipositive if  $g = \prod_j a_j x_j a_j^{-1}$  where  $a_j \in G$  and  $x_j \in X$.   We give a solution for the quasipositivity problem (i.e. we present an algorithm deciding if a given element is quasipositive or not) for a free group with any number of generators and for the group of braids with three strings. In the both cases,  X  is the standard set of generators.

The complexity (the time of work) of our algorithms is   $O(n^{k+1})$   where  n   is the length and   k  is the algebraic length (the exponent sum) of the word.

The result on the free group is not new but we present it here because our proof of this result serves as a model of the proof for 3-braids.

The problem of quasipositivity in braid groups is motivated by the topology of plane real algebraic curves (16th Hilbert's problem). In particular, our result can be interpreted as a classification of trigonal real pseudoholomorphic curves on rational ruled surfaces.