We consider arrow diagrams of links in $$S^3$$S 3 and define k-moves on such diagrams, for any $$k\in \mathbb {N}$$k ∈ N . We study the equivalence classes of links in $$S^3$$S 3 up to k-moves. For $$k=2$$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 $$-1$$- 1 , when k is even. It follows that, for any $$k\ge 5$$k ≥ 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 $$L_{p,1}$$L p , 1 .