Let B be a 3-ball and t= t1∪...∪ tn a union of mutually disjoint n arcs properly embedded in B. Then we call the pair (B, t) an n-string tangle. We say that an n-string tangle (B, t) is trivial if (B, t) is homeomorphic to (D× I,{x1,..., xn}× I) as pairs, where D is a disk and xi is a point in intD (i= 1,..., n). According to [1], we say that (B, t) is essential if cl (∂ B− N (t)) is incompressible and∂-incompressible in cl (B− N (t)). And, according to [4], we say that (B, t) is free if π1 (B− t) is a free group. We note that (B, t) is free if and only if cl (B− N (t)) is a handlebody ([3, 5. 2]).