The unknotting number of a knot is the minimum number of crossing changes necessary to transform the knot into a trivial knot. It's a classical and rather intractable knot invariant. We'll discuss a variation thereof: the minimum number $n$ such that the knot can be transformed into a knot with trivial Alexander polynomial by $n$ positive and $n$ negative crossing changes. We'll see that this knot invariant can equivalently be characterized in terms of the Blanchfield form, and also as minimal genus of certain surfaces in the 4-ball that co-bound the knot. Finally, we'll discuss lower bounds for this invariant coming from the linking pairings of cyclic branched coverings. The talk is based on work in progress with Peter Feller.