The complexity of pinning simple multiloops

Abstract: A multiloop with $s\in \mathbb{N}$ strands is a generic immersion $γ\colon \sqcup_1^s \mathbb{S}^1 \looparrowright Σ$ of the union of $s$ circles into a surface $Σ$, considered up to homeomorphisms. A pinning set of $γ$ is a set of points $P\subset Σ\setminus \operatorname{im}(γ)$, such that in the punctured surface $Σ\setminus P$, the immersion $γ$ has the minimal number of double points in its homotopy class. Its pinning number $\varpi(γ)$ is the minimum cardinal of its pinning sets. In any fixed orientable surface $Σ$, the pinning problem which given a multiloop $γ$ and $k\in \mathbb{N}$ decides whether $\varpi(γ)\le k$ has been show to be NP-complete, even in restrictions to loops (with $s=1$ strand). In this work we study the complexity of the pinning problem in restriction to multiloops whose strands are simple (embedded circles). We show that in any fixed oriented surface $Σ$, the problem is in P when $s\leq 3$ and NP-complete when $s\geq 20$, and present some follow-up questions and conjectures.