  For $p\geq 1$ one can define a generalization of the unknotting number $tu_p$
called the $p$th untwisting number which counts the number of null-homologous
twists on at most $2p$ strands required to convert the knot to the unknot. We
show that for any $p\geq 2$ the difference between the consecutive untwisting
numbers $tu_{p-1}$ and $tu_p$ can be arbitrarily large. We also show that torus
knots exhibit arbitrarily large gaps between $tu_1$ and $tu_2$.
