{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/pin-the-loop-taut-a-one-player-topologame","title":"The pinning ideal of a multiloop","arxiv_id":"2405.16216","date":"2024-05-25","proceeding":null,"authors":["Christopher-Lloyd Simon","Ben Stucky"],"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.","url_abs":"https://arxiv.org/abs/2405.16216v3","url_pdf":"https://arxiv.org/pdf/2405.16216v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"pin-the-loop-taut-a-one-player-topologame","repo_url":"https://github.com/ChristopherLloyd/LooPin","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"pin-the-loop-taut-a-one-player-topologame","repo_url":"https://github.com/christopherlloyd/loopindex","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}