{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:2GTEPBSDV2FLKNIUFDHTVYKPQ5","short_pith_number":"pith:2GTEPBSD","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"},"canonical_sha256":"d1a6478643ae8ab5351428cf3ae14f8773bb66761b5c89f4c979d539ef672c7f","source":{"kind":"arxiv","id":"1709.05118","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.05118","created_at":"2026-05-17T23:51:14Z"},{"alias_kind":"arxiv_version","alias_value":"1709.05118v2","created_at":"2026-05-17T23:51:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05118","created_at":"2026-05-17T23:51:14Z"},{"alias_kind":"pith_short_12","alias_value":"2GTEPBSDV2FL","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"2GTEPBSDV2FLKNIU","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"2GTEPBSD","created_at":"2026-05-18T12:30:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:2GTEPBSDV2FLKNIUFDHTVYKPQ5","target":"record","payload":{"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"},"canonical_sha256":"d1a6478643ae8ab5351428cf3ae14f8773bb66761b5c89f4c979d539ef672c7f","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"},"source_kind":"arxiv","source_id":"1709.05118","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:51:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qkt7IGXrvJ0UEnqws2PzVU14ZUyG1mnY/uqNFtDKjtq2tcoAQcgI1Vh/qNUufACeGGtpy8yJfDH5TUS3qe83Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:28:42.863582Z"},"content_sha256":"0d185d371ff5b3dc26258dbf258a809a48fded18999805a0dff61e4950e861b0","schema_version":"1.0","event_id":"sha256:0d185d371ff5b3dc26258dbf258a809a48fded18999805a0dff61e4950e861b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:2GTEPBSDV2FLKNIUFDHTVYKPQ5","target":"graph","payload":{"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 $c(\\theta_{K_1,K_2}^n)=n(c(K_1)+c(K_2))$. We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05118","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:51:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aYvJMOdshC8VAxcsrrarYDYworZz9R9NDCTRBnB23f8LYELjtwPGSfMsLzQghJeVzBAnLqk2UB/I0tMCippJAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:28:42.863927Z"},"content_sha256":"ac73d8f86d8430105bfa9ae3c1026a0b265462bf610bd5f086def669104a9062","schema_version":"1.0","event_id":"sha256:ac73d8f86d8430105bfa9ae3c1026a0b265462bf610bd5f086def669104a9062"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2GTEPBSDV2FLKNIUFDHTVYKPQ5/bundle.json","state_url":"https://pith.science/pith/2GTEPBSDV2FLKNIUFDHTVYKPQ5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2GTEPBSDV2FLKNIUFDHTVYKPQ5/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T07:28:42Z","links":{"resolver":"https://pith.science/pith/2GTEPBSDV2FLKNIUFDHTVYKPQ5","bundle":"https://pith.science/pith/2GTEPBSDV2FLKNIUFDHTVYKPQ5/bundle.json","state":"https://pith.science/pith/2GTEPBSDV2FLKNIUFDHTVYKPQ5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2GTEPBSDV2FLKNIUFDHTVYKPQ5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:2GTEPBSDV2FLKNIUFDHTVYKPQ5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b06473e16ff9a7a5a7d1a2bdc77d6bc290a847c582a262e3dde2029447b81340","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-09-15T09:10:30Z","title_canon_sha256":"e234dcbab5eca9b809d4d59f7cae95cb1a4eab0b13e646686a15eefcea40f4d1"},"schema_version":"1.0","source":{"id":"1709.05118","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.05118","created_at":"2026-05-17T23:51:14Z"},{"alias_kind":"arxiv_version","alias_value":"1709.05118v2","created_at":"2026-05-17T23:51:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05118","created_at":"2026-05-17T23:51:14Z"},{"alias_kind":"pith_short_12","alias_value":"2GTEPBSDV2FL","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"2GTEPBSDV2FLKNIU","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"2GTEPBSD","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:ac73d8f86d8430105bfa9ae3c1026a0b265462bf610bd5f086def669104a9062","target":"graph","created_at":"2026-05-17T23:51:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $c(\\theta_{K_1,K_2}^n)=n(c(K_1)+c(K_2))$. We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the a","authors_text":"Benjamin Bode","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-09-15T09:10:30Z","title":"Crossing numbers of composite knots and spatial graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05118","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0d185d371ff5b3dc26258dbf258a809a48fded18999805a0dff61e4950e861b0","target":"record","created_at":"2026-05-17T23:51:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b06473e16ff9a7a5a7d1a2bdc77d6bc290a847c582a262e3dde2029447b81340","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-09-15T09:10:30Z","title_canon_sha256":"e234dcbab5eca9b809d4d59f7cae95cb1a4eab0b13e646686a15eefcea40f4d1"},"schema_version":"1.0","source":{"id":"1709.05118","kind":"arxiv","version":2}},"canonical_sha256":"d1a6478643ae8ab5351428cf3ae14f8773bb66761b5c89f4c979d539ef672c7f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d1a6478643ae8ab5351428cf3ae14f8773bb66761b5c89f4c979d539ef672c7f","first_computed_at":"2026-05-17T23:51:14.819496Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:14.819496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"evIRoFjWR4G3BFOmhCFA42uSzBTb2DVLfe1wMsVYjZthdwmjbrt9RvvDX8VFy/D8cSRhBdLGi2UjXbNLIXrZBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:14.819996Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.05118","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0d185d371ff5b3dc26258dbf258a809a48fded18999805a0dff61e4950e861b0","sha256:ac73d8f86d8430105bfa9ae3c1026a0b265462bf610bd5f086def669104a9062"],"state_sha256":"e718905a300defb0de447bd88513f61a66a6d7a7a298d9c3acc35687799c8e01"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mwb6wZYSXH8/1ugiErEsWZNXI/NzK8IwHRTAHAYd4ZUTNM4RBA/mp/Oz2qbLloLDcwhPojOjwPpV3RjLn948Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T07:28:42.865948Z","bundle_sha256":"ecb90c0869513a6ecf3f4b5f2af14cb1ba0221373f73b1cc8e8838c4384866c6"}}