On embeddings of $2$-string tangles into the unknot, the unlink and split links

Abstract

In this paper we study further when tangles embed into the unknot, the unlink or a split link. In particular, we study obstructions to these properties through geometric characterizations, tangle sums and colorings. As an application we determine when each prime 2-string tangle with up to seven crossings embeds into the unknot, the unlink or a split link.