The unknotting number of a knot is the minimum number of crossings one must change
to turn that knot into the unknot. We work with a generalization of the unknotting
number due to Mathieu–Domergue, which we call the untwisting number. The
–untwisting
number is the minimum number (over all diagrams of a knot) of full twists on at
most
strands of a knot, with half of the strands oriented in each direction, necessary to
transform that knot into the unknot. In previous work, we showed that the
unknotting and untwisting numbers can be arbitrarily different. In this paper, we
show that a common route for obstructing low unknotting number, the Montesinos
trick, does not generalize to the untwisting number. However, we use a different
approach to get conditions on the Heegaard Floer correction terms of the branched
double cover of a knot with untwisting number one. This allows us to obstruct several
– and
–crossing knots
from being unknotted by a single positive or negative twist. We also use the Ozsváth–Szabó
invariant and the
Rasmussen
invariant to
differentiate between the
–
and
–untwisting
numbers for certain
.
