{"paper":{"title":"Closed polylines with fixed self-intersection index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","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.","cross_cats":[],"primary_cat":"math.MG","authors_text":"Dmitri Fomin","submitted_at":"2026-05-06T23:06:12Z","abstract_excerpt":"We investigate the existence of closed polylines (also known as closed polygonal chains or self-crossing polygons) that intersect each of their edges the same number of times. 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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We present a complete solution for k = 3 and k = 4, as well as the proof of some non-existence theorems. In conclusion, we show that, for an arbitrary positive integer k, a polyline of the required type exists for any sufficiently large integer n such that nk is even.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The polylines are assumed to be closed chains in the Euclidean plane whose only intersections are transverse crossings, with the self-intersection index per edge being well-defined and independent of the particular embedding chosen.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"81cd943f485a08ab9b52ee0e712ececbbe07cf182c36d7fcb77cce45e8c3a04b"},"source":{"id":"2605.05506","kind":"arxiv","version":3},"verdict":{"id":"d14d8a04-a9f8-4764-9661-bb79c1fe647f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-11T01:00:12.407840Z","strongest_claim":"We present a complete solution for k = 3 and k = 4, as well as the proof of some non-existence theorems. In conclusion, we show that, for an arbitrary positive integer k, a polyline of the required type exists for any sufficiently large integer n such that nk is even.","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","weakest_assumption":"The polylines are assumed to be closed chains in the Euclidean plane whose only intersections are transverse crossings, with the self-intersection index per edge being well-defined and independent of the particular embedding chosen.","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."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.05506/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T20:31:19.523132Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T13:29:42.732978Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"d27d14395b799e4145da13ce44d8e9957354ecdcb5eff9ca7d51bedb01e41e9a"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"496d8d345f84d8f0d469c924897abde1a6bc2093d3ab8d6dcc8aa452c2c6c4c5"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}