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The most general question in this corner of combinatorial geometry asks for all pairs $(n, k)$ such that there exists a closed polyline with $n$ edges, each intersecting the same polyline exactly $k$ times. For $k = 1$ and $k = 2$, this is a very simple question answered several decades ago. In this article, we present a complete solution for $k = 3, 4, 6$, as well as the proof of some non-existence theorems. In conclusi","authors_text":"Dmitri Fomin","cross_cats":[],"headline":"Closed polylines exist in which every one of the n edges is crossed exactly k times, for every k and all sufficiently large n making nk even.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-05-06T23:06:12Z","title":"Closed polylines with fixed self-intersection index"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.05506","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-11T01:00:12.407840Z","id":"d14d8a04-a9f8-4764-9661-bb79c1fe647f","model_set":{"reader":"grok-4.3"},"one_line_summary":"Complete solutions for uniform self-intersection index k=3 and k=4, plus a general existence theorem for sufficiently large n when nk is even.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Closed polylines exist in which every one of the n edges is crossed exactly k times, for every k and all sufficiently large n making nk even.","strongest_claim":"We present a complete solution for k = 3 and k = 4, as well as the proof of some non-existence theorems. 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