# 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.

## Key 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 $L_{p,1}$.

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Source: https://tomesphere.com/paper/1908.00342