Abstract
We consider arrow diagrams of links in S^3 and define k-moves on such diagrams, for any kin 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 kge 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}.
Highlights
Several types of moves on links in S3 were studied extensively for a long time
We introduce k-moves, k ∈ N, on links in S3. They are defined with the help of arrow diagrams of links
For k ≥ 5, we show that the value of the Jones polynomial in a k-th primitive root of unity is unchanged under a k-move, if k is odd, and changes sign, if k is even
Summary
Several types of moves on links in S3 were studied extensively for a long time (see for example [4,13]). For k = 1, all links are related by k-moves. For k = 2, all knots (but not all links) are related by k-moves. For k ≥ 5, we show that the value of the Jones polynomial in a k-th primitive root of unity is unchanged under a k-move, if k is odd, and changes sign, if k is even (see Theorem 1). It follows that there are infinitely many classes of knots modulo k-moves when k ≥ 5
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