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Conversely, using the deleted product cohomology, we show that for $K_5$ and $K_{3,3}$, if $A$ is any set of pairs of independent edges, and $A$ has odd cardinality, then there is a drawing in the plane for which each element in $A$ cross an odd number of times, while each pair of independent edges not in $A$ cross an even number of times. For $K_6$ we show that there is a drawing wit"},"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":"1903.06292","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-14T23:03:08Z","cross_cats_sorted":[],"title_canon_sha256":"9765643a24964a49fa1bd3f358d257e38505d615c4124dba0bb42afa65525e44","abstract_canon_sha256":"d5b8b39982a77d8652d1c1295175ad59c7fa2c2826adb7ed81bf5e57a8907afc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:11.209658Z","signature_b64":"8uKLnHTf5zPSfU08Dm9o+CtOJJGZTLsJ9ySoSopQoNMVJ+T71gT1FHZfZ2EsaVQcQvgPlzBmh2bfXxUba41ODA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6bf32db24d21cda18339abffae5c8ce9ab7bd21f8a72ae9e7881baac9445f1af","last_reissued_at":"2026-05-17T23:51:11.209065Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:11.209065Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bad drawings of small complete graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Emily Groves, Grant Cairns, Yuri Nikolayevsky","submitted_at":"2019-03-14T23:03:08Z","abstract_excerpt":"We show that for $K_5$ (resp.~ $K_{3,3}$) there is a drawing with $i$ independent crossings, and no pair of independent edges cross more than once, provided $i$ is odd with $1\\le i\\le 15$ (resp.~ $1\\le i\\le 17$). 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