Papers › The pinning ideal of a multiloop
The pinning ideal of a multiloop
Christopher-Lloyd Simon, Ben Stucky
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A multiloop γ:⊔₁ˢ 𝕊¹ ↬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⊂𝔽∖im(γ), such that in the punctured surface 𝔽 ∖P, the immersion γ has the minimal number of double points in its homotopy class. The collection of pinning sets of γ forms a poset under inclusion called the pinning ideal 𝒫ℐ(γ) which is endowed with the cardinal function whose minimum defines the pinning number ϖ(γ). 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 ≈1000 of the smallest multiloops in the sphere, available in the online catalog LooPindex.
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