# The splitting genus of Alexander split links

### Abstract

We say 2-component link L in $\mathbb{S}^3$ is “Alexander split” if its Alexander polynomial is zero. It turns out that L is Alexander split exactly when the maximal abelian cover of its exterior has non-zero $H_2$. In fact, with its natural module structure, this $H_2$ has rank 1. We define the splitting genus of L to be the minimal genus of surfaces representing a generator. I’ll discuss the development of this invariant, fundamental results, and potential applications. Parts are joint work with Chris Anderson.

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