Fundamental group in the projective knot theory

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.