  The untwisting number of a knot K is the minimum number of null-homologous
twists required to convert K to the unknot. Such a twist can be viewed as a
generalization of a crossing change, since a classical crossing change can be
effected by a null-homologous twist on 2 strands. While the unknotting number
gives an upper bound on the smooth 4-genus, the untwisting number gives an
upper bound on the topological 4-genus. The surgery description number, which
allows multiple null-homologous twists in a single twisting region to count as
one operation, lies between the topological 4-genus and the untwisting number.
We show that the untwisting and surgery description numbers are different for
infinitely many knots, though we also find that the untwisting number is at
most twice the surgery description number plus 1.
