On knots and links in lens spaces

TL;DR

This paper explores the properties of knots and links in lens spaces, introducing diagrammatic methods, Reidemeister moves, and algebraic invariants like Alexander polynomials to understand their topology.

Contribution

It provides a complete set of Reidemeister moves, a Wirtinger presentation, and methods to compute homology and Alexander polynomials for links in lens spaces.

Findings

Complete finite set of Reidemeister moves for lens space links

Wirtinger presentation for the fundamental group

Methods to compute homology and Alexander polynomials

Abstract

In this paper we study some aspects of knots and links in lens spaces. Namely, if we consider lens spaces as quotient of the unit ball $B^{3}$ with suitable identification of boundary points, then we can project the links on the equatorial disk of $B^{3}$, obtaining a regular diagram for them. In this contest, we obtain a complete finite set of Reidemeister type moves establishing equivalence, up to ambient isotopy, a Wirtinger type presentation for the fundamental group of the complement of the link and a diagrammatic method giving the first homology group. We also compute Alexander polynomial and twisted Alexander polynomials of this class of links, showing their correlation with Reidemeister torsion.