Fundamental groups in projective knot theory

Abstract

We relate properties of a link $L$ in the projective space $\mathbb{R}P^3$ to properties of the group $\pi_1(\mathbb{R}P^3 \smallsetminus L)$: \begin{itemize} \item $L$ is isotopic to a projective line if and only if $\pi_1(\mathbb{R}P^3 \smallsetminus L) = \mathbb{Z}$. \item $L$ is isotopic to an affine circle if and only if $\pi_1(\mathbb{R}P^3 \smallsetminus L) = \mathbb{Z} \ast \mathbb{Z}_{/2}$. \item $L$ 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. \end{itemize} 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.