# Fundamental groups in projective knot theory

### Julia Viro

Stony Brook University, USA### Oleg Viro

Stony Brook University, USA

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## Abstract

We relate properties of a link $L$ in the projective space $RP_{3}$ to properties of the group $π_{1}(RP_{3}∖L)$: \begin{itemize} \item $L$ is isotopic to a projective line if and only if $π_{1}(RP_{3}∖L)=Z$. \item $L$ is isotopic to an affine circle if and only if $π_{1}(RP_{3}∖L)=Z∗Z_{/2}$. \item $L$ is isotopic to a link disjoint from a projective plane if and only if $π_{1}(RP_{3}∖L)$ contains a non-trivial element of order two. \end{itemize} A simple algorithm which finds a system of generators and relations for $π_{1}(RP_{3}∖L)$ in terms of a link diagram of $L$ is provided.