Narrow Results

An H(2)-move is an unknotting operation of a knot, which is performed by adding a half-twisted band. We define the H(2)-Gordian distance of two knots to be the minimum number of H(2)-moves needed to transform one into the other. We give several methods to estimate the H(2)-Gordian distance of knots. Then we give a table of H(2)-Gordian distances of knots with up to 7 crossings.

We consider an $H(2)$-move between two torus links of type $(2,2n)$, $T_{2n}$. We give some necessary conditions for two links $T_{2n}$ and $T_{2m}$ which are related by a single $H(2)$-move. In particular, we show that $T_{2n}$ is obtained by a single $H(2)$-move from the Hopf link if and only if $n=\pm 1$, $\pm 3$, and $T_{2n}$ is obtained by a single $H(2)$-move from the trivial 2-component link if and only if $n=0$, $\pm 2$,