ON TANGLE DECOMPOSITIONS OF TUNNEL NUMBER ONE LINKS

We will discuss tangle decompositions of tunnel number one links, and show that any tangle composite tunnel number one link can be obtained as the union of a quasi Hopf tangle and a trivial tangle. We also consider the uniqueness of tangle decompositions of such links, and show that there are infinitely many tunnel number one links that admit non-isotopic essential tangle decompositions.