Links with splitting number one

The unknotting number of a knot is an incredibly difficult invariant to compute. In fact, there are many knots which are conjectured to have unknotting number 2 but for which no proof of this is currently available. It therefore remains an unsolved problem to find an algorithm that determines whether a knot has unknotting number one. In my talk, I will show that an analogous problem for links is soluble. We say that a link has splitting number one if some crossing change turns it into a split link. I will give an algorithm that determines whether a link has splitting number one. (In the case where the link has two components, we must make a hypothesis on their linking number.) The proof that the algorithm works uses sutured manifolds and normal surfaces.