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 ${\pi }_1(\mathbb{R}{P}^3\setminus L)$: \begin{itemize} \item $L$ is isotopic to a projective line if and only if ${\pi }_1(\mathbb{R}{P}^3\setminus L)=\mathbb{Z}$. \item $L$ is isotopic to an affine circle if and only if ${\pi }_1(\mathbb{R}{P}^3\setminus L)=\mathbb{Z}\ast {\mathbb{Z}}_{/2}$. \item $L$ is isotopic to a link disjoint from a projective plane if and only if ${\pi }_1(\mathbb{R}{P}^3\setminus L)$ contains a non-trivial element of order two. \end{itemize} A simple algorithm which finds a system of generators and relations for ${\pi }_1(\mathbb{R}{P}^3\setminus L)$ in terms of a link diagram of $L$ is provided.