Fundamental group in the projective knot theory
2020
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.
Keywords:
- Correction
- Source
- Cite
- Save
- Machine Reading By IdeaReader
10
References
0
Citations
NaN
KQI