Projective Braids in $\mathbb{R}P^3$

Abstract

Links in projective space $\mathbb{R}P^3$ can be represented by diagrams in $\mathbb{R}P^2$ where the projective plane $\mathbb{R}P^2$ is represented by a disk with antipodal identifications on the boundary. In this context if we place an $n-$strand braid $β$ on this disk keeping its end points on the boundary then due to the identification of antipodal points it naturally represents a link diagram in $\mathbb{R}P^3$. We call this the projective closure of the braid $β$. In this paper we show that not all links in $\mathbb{R}P^3$ possess a diagram represented by projective closure of some braid.