The unknotting number of a knot is the minimum number of crossings one must
change to turn that knot into the unknot. The algebraic unknotting number is the
minimum number of crossing changes needed to transform a knot into an Alexander
polynomial-one knot. We work with a generalization of unknotting number due to
Mathieu and Domergue, which we call the untwisting number. The untwisting
number is the minimum number (over all diagrams of a knot) of right- or
left-handed twists on even numbers of strands of a knot, with half of the strands
oriented in each direction, necessary to transform that knot into the unknot.
We show that the algebraic untwisting number is equal to the algebraic
unknotting number. However, we also exhibit several families of knots for
which the difference between the unknotting and untwisting numbers is
arbitrarily large, even when we only allow twists on a fixed number of strands or
fewer.
