Let L be a link in S^3 of two components. We say L is `Alexander split' if its Alexander polynomial is zero. It turns out that this is equivalent to the second homology of the universal abelian cover of the exterior X of L having rank 1 as a Z[H_1(X)] module. As a generator is represented by a closed connected orientable surface, we define the `splitting genus' of an Alexander split link to be the minimum genus among such surfaces. In addition to defining this invariant, we discuss a few fundamental results, other related invariants, and directions for future research, Parts of this are joint with Chris Anderson.
Department of Mathematics