Fundamental group in the projective knot theory

TL;DR

This paper explores the relationship between the fundamental group of links in real projective 3-space and their isotopy classes, providing characterizations and an algorithm for computing the group from a link diagram.

Contribution

It establishes new characterizations of links in projective space based on their fundamental groups and introduces an algorithm to compute these groups from diagrams.

Findings

Characterization of links isotopic to a projective line via fundamental group

Identification of links isotopic to affine circles through group structure

Algorithm for computing the fundamental group from a link diagram

Abstract

In this paper, properties of a link $L$ in the projective space $\mathbb R P^3$ are related to properties of its group $\pi_1(\mathbb R P^3\smallsetminus L)$: $L$ is isotopic to a projective line if and only if $\pi_1(\mathbb R P^3\smallsetminus L)=\mathbb Z$. $L$ is isotopic to an affine circle if and only if $\pi_1(\mathbb R P^3\smallsetminus L)=\mathbb Z*\mathbb Z_{/2}$. $L$ is isotopic to a link disjoint from a projective plane if and only if $\pi_1(\mathbb R P^3\smallsetminus L)$ contains a non-trivial element of order two. A simple algorithm which finds a system of generators and relations for $\pi_1(\mathbb R P^3\smallsetminus L)$ in terms of a link diagram of $L$ is provided.