Gaps between consecutive untwisting numbers

Abstract

For $p\ge 1$ one can define a generalization of the unknotting number $t{u}_p$ 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 $p\ge 2$ the difference between the consecutive untwisting numbers $t{u}_{p-1}$ and $t{u}_p$ can be arbitrarily large. We also show that torus knots exhibit arbitrarily large gaps between $t{u}_1$ and $t{u}_2$.

6 pages, 4 figures