  We propose a simple geometrical construction of topological invariants of
3-strand Brownian braids viewed as world lines of 3 particles performing
independent Brownian motions in the complex plane z. Our construction is based
on the properties of conformal maps of doubly-punctured plane z to the
universal covering surface. The special attention is paid to the case of
indistinguishable particles. Our method of conformal maps allows us to
investigate the statistical properties of the topological complexity of a bunch
of 3-strand Brownian braids and to compute the expectation value of the
irreducible braid length in the non-Abelian case.
