A tangle T is a pair formed by a ball and collection of properly embedded disjoint arcs in B. Tangles are useful for decomposing links and have presence in applications in science. In this talk we will discuss the case where the unknot and split links have decompositions with a given tangle T as factor. As an application we obtain a complete classification for tangles up to seven crossings with such properties. 

This is joint work with António Salgueiro.