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We prove that for large enough $n$ we have $c(\\theta_{K_1,K_2}^n)=n(c(K_1)+c(K_2))$. We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the a"},"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":"1709.05118","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-09-15T09:10:30Z","cross_cats_sorted":[],"title_canon_sha256":"e234dcbab5eca9b809d4d59f7cae95cb1a4eab0b13e646686a15eefcea40f4d1","abstract_canon_sha256":"b06473e16ff9a7a5a7d1a2bdc77d6bc290a847c582a262e3dde2029447b81340"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:14.819996Z","signature_b64":"evIRoFjWR4G3BFOmhCFA42uSzBTb2DVLfe1wMsVYjZthdwmjbrt9RvvDX8VFy/D8cSRhBdLGi2UjXbNLIXrZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1a6478643ae8ab5351428cf3ae14f8773bb66761b5c89f4c979d539ef672c7f","last_reissued_at":"2026-05-17T23:51:14.819496Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:14.819496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Crossing numbers of composite knots and spatial graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Benjamin Bode","submitted_at":"2017-09-15T09:10:30Z","abstract_excerpt":"We study the minimal crossing number $c(K_{1}\\# K_{2})$ of composite knots $K_{1}\\# K_{2}$, where $K_1$ and $K_2$ are prime, by relating it to the minimal crossing number of spatial graphs, in particular the $2n$-theta curve $\\theta_{K_{1},K_{2}}^n$ that results from tying $n$ of the edges of the planar embedding of the $2n$-theta graph into $K_1$ and the remaining $n$ edges into $K_2$. We prove that for large enough $n$ we have $c(\\theta_{K_1,K_2}^n)=n(c(K_1)+c(K_2))$. 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