Fundamental group and twisted Alexander polynomial of link complement in 3-torus

TL;DR

This paper develops a diagrammatic method to analyze knots and links in the 3-torus, providing presentations for their fundamental groups and homology, and computing their Alexander and twisted Alexander polynomials.

Contribution

It introduces a finite set of Reidemeister moves for links in T^3 and computes their fundamental groups and Alexander polynomials, advancing knot theory in 3-torus environments.

Findings

Finite Reidemeister moves for links in T^3

Presentations for fundamental groups and homology groups

Explicit calculations of Alexander and twisted Alexander polynomials

Abstract

We consider a diagrammatic approach to investigate tame knots and links in three dimensional torus $T^3$. We obtain a finite set of generalised Reidemeister moves for equivalent links up to ambient isotopy. We give a presentation for fundamental group of link complement in 3-torus $T^3$ and the first homology group. We also compute Alexander polynomial and twisted Alexander polynomials of this class of links.