On parallelism of strings in tangles

We consider when two proper arcs t1 and t2 in a ball or handlebody V are parallel. A necessary and sufficient condition of t1 and t2 being parallel given in this paper is as follows: V − int N(t1 ∪ t2) ≅ (V − int N(ti)) ♮ (a solid torus) for i = 1 and 2, where ♮ denotes boundary connected sum.