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Generalizing the result by \\'{A}brego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by $x$-monotone curves. In fact, our proof shows that the conjecture remains true for $x$-monotone drawings of $K_n$ in which adjacent edges may cross an even number of times, and instead of the"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1312.3679","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-13T00:00:39Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"cd51721d362a1c434184364c0c443701be99739cc275856a8866c7a1e2b52c5e","abstract_canon_sha256":"f2b93977eaeb4030f59b3ba50dd259ba8049d9f035304bea21c27d94490191ee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:23.233528Z","signature_b64":"A/7iM2RQl+ywvF1y+9Fn6FPqNuDORQFE1vmECMSiApTt7kzrQPLnWRe5Jy9nf6Ok1sOhPwZvd8AHTJ24Dn9MDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a0423ec94eef98b370f30262d5d0d4438f6afe76edc3471c5230fec48fa4683","last_reissued_at":"2026-05-18T02:28:23.232908Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:23.232908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Crossing numbers and combinatorial characterization of monotone drawings of $K_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jan Kyn\\v{c}l, Martin Balko, Radoslav Fulek","submitted_at":"2013-12-13T00:00:39Z","abstract_excerpt":"In 1958, Hill conjectured that the minimum number of crossings in a drawing of $K_n$ is exactly $Z(n) = \\frac{1}{4} \\lfloor\\frac{n}{2}\\rfloor \\left\\lfloor\\frac{n-1}{2}\\right\\rfloor \\left\\lfloor\\frac{n-2}{2}\\right\\rfloor\\left\\lfloor\\frac{n-3}{2}\\right\\rfloor$. Generalizing the result by \\'{A}brego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by $x$-monotone curves. In fact, our proof shows that the conjecture remains true for $x$-monotone drawings of $K_n$ in which adjacent edges may cross an even number of times, and instead of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3679","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1312.3679","created_at":"2026-05-18T02:28:23.233031+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.3679v3","created_at":"2026-05-18T02:28:23.233031+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.3679","created_at":"2026-05-18T02:28:23.233031+00:00"},{"alias_kind":"pith_short_12","alias_value":"NICCH3EU534Y","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_16","alias_value":"NICCH3EU534YWNYP","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_8","alias_value":"NICCH3EU","created_at":"2026-05-18T12:27:52.871228+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NICCH3EU534YWNYPGATC2XINIQ","json":"https://pith.science/pith/NICCH3EU534YWNYPGATC2XINIQ.json","graph_json":"https://pith.science/api/pith-number/NICCH3EU534YWNYPGATC2XINIQ/graph.json","events_json":"https://pith.science/api/pith-number/NICCH3EU534YWNYPGATC2XINIQ/events.json","paper":"https://pith.science/paper/NICCH3EU"},"agent_actions":{"view_html":"https://pith.science/pith/NICCH3EU534YWNYPGATC2XINIQ","download_json":"https://pith.science/pith/NICCH3EU534YWNYPGATC2XINIQ.json","view_paper":"https://pith.science/paper/NICCH3EU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.3679&json=true","fetch_graph":"https://pith.science/api/pith-number/NICCH3EU534YWNYPGATC2XINIQ/graph.json","fetch_events":"https://pith.science/api/pith-number/NICCH3EU534YWNYPGATC2XINIQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NICCH3EU534YWNYPGATC2XINIQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NICCH3EU534YWNYPGATC2XINIQ/action/storage_attestation","attest_author":"https://pith.science/pith/NICCH3EU534YWNYPGATC2XINIQ/action/author_attestation","sign_citation":"https://pith.science/pith/NICCH3EU534YWNYPGATC2XINIQ/action/citation_signature","submit_replication":"https://pith.science/pith/NICCH3EU534YWNYPGATC2XINIQ/action/replication_record"}},"created_at":"2026-05-18T02:28:23.233031+00:00","updated_at":"2026-05-18T02:28:23.233031+00:00"}