Linking numbers of pseudo-branch curves

Let $K$ be a knot in $S^3$ and $f\colon M \to S^3$ a cover branched along $K$. Under certain hypotheses, the linking numbers in $M$ between the components of $f^{-1}(K)$ are an invariant of $K$. This invariant was crucial for expanding the knot table to include knots of more than 8 crossings, among other uses. Also important but less well-studied are linking numbers between ``pseudo-branch curves", or lifts to $M$ of simple closed curves in the complement of $K$. I describe a method for computing such linking numbers. I will also explain the motivation for this work, and how it can be used in the classification of branched covers between four-manifolds with singular branching sets. Joint with Patricia Cahn.