On some moves on links and the Hopf crossing number

TL;DR

This paper investigates the effects of k-moves on links in 3-spheres, revealing their impact on link equivalence, polynomial invariants, and the Hopf crossing number, with implications for links in lens spaces.

Contribution

It introduces the concept of k-moves on links, analyzes their effects on link equivalence and invariants, and explores their relation to the Hopf crossing number and lens space links.

Findings

Any two knots are equivalent under 2-moves.

Jones polynomial at k-th roots of unity is invariant or changes predictably under k-moves.

The Hopf crossing number can be unbounded for certain knot families.

Abstract

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 $L_{p,1}$.