{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:QXSSPY3ANQ3GDLQXKY47TFB4MB","short_pith_number":"pith:QXSSPY3A","canonical_record":{"source":{"id":"1112.3693","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-12-16T00:32:47Z","cross_cats_sorted":[],"title_canon_sha256":"7171c6a243fde34f5da2ebf7112a75d133ffbeb09a18b1f2d856238ffdf0e66e","abstract_canon_sha256":"e1793c91b05ca65845bc3743ab74b049a721f7a90c9b003e0dc8a4247f298fef"},"schema_version":"1.0"},"canonical_sha256":"85e527e3606c3661ae175639f9943c6071446c8bca61f00ea757310022f777f7","source":{"kind":"arxiv","id":"1112.3693","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.3693","created_at":"2026-05-18T03:53:54Z"},{"alias_kind":"arxiv_version","alias_value":"1112.3693v2","created_at":"2026-05-18T03:53:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.3693","created_at":"2026-05-18T03:53:54Z"},{"alias_kind":"pith_short_12","alias_value":"QXSSPY3ANQ3G","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QXSSPY3ANQ3GDLQX","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QXSSPY3A","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:QXSSPY3ANQ3GDLQXKY47TFB4MB","target":"record","payload":{"canonical_record":{"source":{"id":"1112.3693","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-12-16T00:32:47Z","cross_cats_sorted":[],"title_canon_sha256":"7171c6a243fde34f5da2ebf7112a75d133ffbeb09a18b1f2d856238ffdf0e66e","abstract_canon_sha256":"e1793c91b05ca65845bc3743ab74b049a721f7a90c9b003e0dc8a4247f298fef"},"schema_version":"1.0"},"canonical_sha256":"85e527e3606c3661ae175639f9943c6071446c8bca61f00ea757310022f777f7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:54.542022Z","signature_b64":"EaCs631Gt/XOqCjeDkrBOv9s+77ppLAcCREUghPW9PsHi/6ptmEZEXinEIIPSbqGM9iv1rF4xm0BzK/RQD5cDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85e527e3606c3661ae175639f9943c6071446c8bca61f00ea757310022f777f7","last_reissued_at":"2026-05-18T03:53:54.541407Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:54.541407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.3693","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":"2026-05-18T03:53:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LxbRIblL0r6NQBUWfuZ2ABkQHyIpDQ3fPIc+VeKRjLZYstoCFkRHO50mmV4WvHQxN1X59tMRpjO6ToDf4WAIBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:39:23.772893Z"},"content_sha256":"086f18da5e420836e92c781994634c0417147304a3990add57ce898479a9c47c","schema_version":"1.0","event_id":"sha256:086f18da5e420836e92c781994634c0417147304a3990add57ce898479a9c47c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:QXSSPY3ANQ3GDLQXKY47TFB4MB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Normal Tori in $\\sharp_n (S^2\\times S^1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Funda G\\\"ultepe","submitted_at":"2011-12-16T00:32:47Z","abstract_excerpt":"The fundamental group of $M = \\sharp_n (S^2\\times S^1)$ is $F_n$, the free group with $n$ generators. There is a 1-1 correspondence between the equivalence classes of $\\mathbb{Z}$-- splittings of $F_n$ and homotopy classes of embedded essential tori in $M$. We define and prove a local notion of minimal intersection of a torus with respect to a maximal sphere system in $M$, which generalizes Hatcher's work \\cite{H1} on 2-spheres in the same manifold."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3693","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":"2026-05-18T03:53:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J9WVn8Wh3xwfAa4mIiHMF6DIhAP161fd5o8IUX6O7hueNOAUfkV3PB6SkZaDDr9pFyeTqRof4WqLy2avB+PbBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:39:23.773383Z"},"content_sha256":"65657e3bc9db1e0df33d412a34258d83e51d0a7fe5eee800000e7b15f4d67d13","schema_version":"1.0","event_id":"sha256:65657e3bc9db1e0df33d412a34258d83e51d0a7fe5eee800000e7b15f4d67d13"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QXSSPY3ANQ3GDLQXKY47TFB4MB/bundle.json","state_url":"https://pith.science/pith/QXSSPY3ANQ3GDLQXKY47TFB4MB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QXSSPY3ANQ3GDLQXKY47TFB4MB/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":"2026-05-27T05:39:23Z","links":{"resolver":"https://pith.science/pith/QXSSPY3ANQ3GDLQXKY47TFB4MB","bundle":"https://pith.science/pith/QXSSPY3ANQ3GDLQXKY47TFB4MB/bundle.json","state":"https://pith.science/pith/QXSSPY3ANQ3GDLQXKY47TFB4MB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QXSSPY3ANQ3GDLQXKY47TFB4MB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:QXSSPY3ANQ3GDLQXKY47TFB4MB","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":"e1793c91b05ca65845bc3743ab74b049a721f7a90c9b003e0dc8a4247f298fef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-12-16T00:32:47Z","title_canon_sha256":"7171c6a243fde34f5da2ebf7112a75d133ffbeb09a18b1f2d856238ffdf0e66e"},"schema_version":"1.0","source":{"id":"1112.3693","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.3693","created_at":"2026-05-18T03:53:54Z"},{"alias_kind":"arxiv_version","alias_value":"1112.3693v2","created_at":"2026-05-18T03:53:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.3693","created_at":"2026-05-18T03:53:54Z"},{"alias_kind":"pith_short_12","alias_value":"QXSSPY3ANQ3G","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QXSSPY3ANQ3GDLQX","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QXSSPY3A","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:65657e3bc9db1e0df33d412a34258d83e51d0a7fe5eee800000e7b15f4d67d13","target":"graph","created_at":"2026-05-18T03:53:54Z","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":"The fundamental group of $M = \\sharp_n (S^2\\times S^1)$ is $F_n$, the free group with $n$ generators. There is a 1-1 correspondence between the equivalence classes of $\\mathbb{Z}$-- splittings of $F_n$ and homotopy classes of embedded essential tori in $M$. We define and prove a local notion of minimal intersection of a torus with respect to a maximal sphere system in $M$, which generalizes Hatcher's work \\cite{H1} on 2-spheres in the same manifold.","authors_text":"Funda G\\\"ultepe","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-12-16T00:32:47Z","title":"Normal Tori in $\\sharp_n (S^2\\times S^1)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3693","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":"sha256:086f18da5e420836e92c781994634c0417147304a3990add57ce898479a9c47c","target":"record","created_at":"2026-05-18T03:53:54Z","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":"e1793c91b05ca65845bc3743ab74b049a721f7a90c9b003e0dc8a4247f298fef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-12-16T00:32:47Z","title_canon_sha256":"7171c6a243fde34f5da2ebf7112a75d133ffbeb09a18b1f2d856238ffdf0e66e"},"schema_version":"1.0","source":{"id":"1112.3693","kind":"arxiv","version":2}},"canonical_sha256":"85e527e3606c3661ae175639f9943c6071446c8bca61f00ea757310022f777f7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"85e527e3606c3661ae175639f9943c6071446c8bca61f00ea757310022f777f7","first_computed_at":"2026-05-18T03:53:54.541407Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:53:54.541407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EaCs631Gt/XOqCjeDkrBOv9s+77ppLAcCREUghPW9PsHi/6ptmEZEXinEIIPSbqGM9iv1rF4xm0BzK/RQD5cDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:53:54.542022Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.3693","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:086f18da5e420836e92c781994634c0417147304a3990add57ce898479a9c47c","sha256:65657e3bc9db1e0df33d412a34258d83e51d0a7fe5eee800000e7b15f4d67d13"],"state_sha256":"4daf82b743e2733d5cbacf8012281f862c619af1c1e724dea7b49fe1ba07851c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vTLu7lH/5KZh5qEpwqkoCK+2884cRYEx5p5Pld9IglkfmuRVCAWgH8u6o6IvTSUxC1KvQzTCyfwsbHewSqzCCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T05:39:23.776483Z","bundle_sha256":"e9ba675bf69a4773baf0386076c99c6f22a0f0b9d1bec5fb8ba69e8aa56bef93"}}