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 ${\pi }_1(\mathbb{R}{P}^3\setminus L)$: $L$ is isotopic to a projective line if and only if ${\pi }_1(\mathbb{R}{P}^3\setminus L)=\mathbb{Z}$. $L$ is isotopic to an affine circle if and only if ${\pi }_1(\mathbb{R}{P}^3\setminus L)=\mathbb{Z}\ast {\mathbb{Z}}_{/2}$. $L$ 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. 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.

21 pages. This is an expanded version of a preprint arXiv:1901.07686 [math.GT] of the second author

References in corpus (2)

• Configurations of skew lines
• Characterization of affine links in the projective space