# Diffeomorphic vs isotopic links in lens spaces

## TL;DR

This paper explores the relationship between diffeomorphic and isotopic links in lens spaces, providing new diagrammatic moves for diffeomorphic links and analyzing how lifts to the 3-sphere determine knot classes.

## Contribution

It introduces a set of moves on diagrams that connect diffeomorphic links in lens spaces and relates these to lifts in the 3-sphere, clarifying the classification of knots.

## Key findings

- Up to four isotopy classes per diffeomorphism class.
- Lift in the 3-sphere determines the knot class for primitive-homologous knots.
- Only four knots share the same lift in the case studied.

## Abstract

Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper we provide a set of moves on disk, band and grid diagrams that connects diffeo equivalent links: there are up to four isotopy equivalent links in each diffeo equivalence class. Moreover, we investigate how the diffeo equivalence relates to the lift of the link in the $3$-sphere: in the particular case of oriented primitive-homologous knots, the lift completely determines the knot class in $L(p,q)$ up to diffeo equivalence, and thus only four possible knots up to isotopy equivalence can have the same lift.