  In this paper we prove certain Hurwitz equivalence properties of $B_n$. In
particular we prove that for $n=3$ every two Artin's factorizations of $\Delta
_3 ^2$ of the form $H_{i_1} ... H_{i_6}, \quad F_{j_1} ... F_{j_6}$ (with $i_k,
j_k \in \{1,2 \}$) 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 are not deformations of each other (Manetti
example). We are constructing a new invariant based on Hurwitz equivalence
class of factorization, to distinguish among diffeomorphic surfaces which are
not deformation of each other.
