Hurwitz equivalence in the braid group B3

In this paper we prove certain Hurwitz equivalence properties of B(n). In particular we prove that for n = 3 every two Artin factorizations of Delta(3)(2) of the form Hi(1) (...) Hi(6), Fj(1) (...) Fj(6) (with i(k), j(k) is an element of {1, 2}) where {H(1), H(2)}, {F(1), F(2)} are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but axe not deformations of each other (Manetti's example). We construct a new invariant based on Hurwitz equivalence of factorizations, to distinguish among diffeomorphic surfaces which are not deformation of each other. The main result of this paper will help us to compute the new invariant.