Plat closures of spherical braids in $\mathbb{R}P^3$

Abstract: We define plat closure for spherical braids to obtain links in $\mathbb{R}P^3$ and prove that all links in $\mathbb{R}P^3$ can be realized in this manner. Given a spherical braid $\beta$ of $2n$ strands in $\mathbb{R}P^3$ we associate a permutation $h_{\beta}$ on $n$ elements called \textit{residual permutation}. We prove that the number of components of the plat closure link of a spherical braid $\beta$ is same as the number of disjoint cycles in $h_{\beta}$. We also present a set of moves on spherical braids in the same spirit as the classical Markov moves on braids. The completeness of this set of moves to capture the entire isotopy classes of the plat closure links is still to be explored.