The pinning ideal of a multiloop

Abstract

A multiloop $\gamma\colon \sqcup_1^s \mathbb{S}^1 \looparrowright \mathbb{F}$ is a generic immersion of a finite union of circles into an oriented surface, considered up to homeomorphisms. A pinning set is a set of points $P\subset \mathbb{F}\setminus \operatorname{im}(\gamma)$, such that in the punctured surface $\mathbb{F} \setminus P$, the immersion $\gamma$ has the minimal number of double points in its homotopy class. The collection of pinning sets of $\gamma$ forms a poset under inclusion called the pinning ideal $\mathcal{PI}(\gamma)$ which is endowed with the cardinal function whose minimum defines the pinning number $\varpi(\gamma)$. We show that the decision problem associated to computing the pinning number of a multiloop is \textsf{NP}-complete, even for loops in the sphere. We give two proofs that it is \textsf{NP}: First, we implement a polynomial algorithm to check if a point-set is pinning, adapting methods of Birman--Series and Cohen--Lustig for computing intersection numbers of curves in surfaces. Second, for loops in the sphere we reduce the problem in polynomial time to a variant of boolean satisfiability by applying a theorem of Hass--Scott characterizing taut loops, and adapting algorithms of Blank and Shor--Van Wyk which decide when a curve in the plane bounds an immersed disc. To show that it is \textsf{NP}-hard we reduce the vertex cover problem for graphs to the pinning problem for plane loops. We use our algorithms to compute the pinning ideals for $\approx 1000$ of the smallest multiloops in the sphere, available in the online catalog LooPindex.