  We consider arrow diagrams of links in $S^3$ and define $k$-moves on such
diagrams, for any $k\in\mathbb N$. We study the equivalence classes of links in
$S^3$ up to $k$-moves. For $k=2$, we show that any two knots are equivalent,
whereas it is not true for links. We show that the Jones polynomial at a $k$-th
primitive root of unity is unchanged by a $k$-move, when $k$ is odd. It is
multiplied by $-1$, when $k$ is even. It follows that, for any $k\ge 5$, there
are infinitely many classes of knots modulo $k$-moves. We use these results to
study the Hopf crossing number. In particular, we show that it is unbounded for
some families of knots. We also interpret $k$-moves as some identifications
between links in different lens spaces $L_{p,1}$.
