{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ASQ6AKS5NS2QVIZ72M4IM6LF47","short_pith_number":"pith:ASQ6AKS5","canonical_record":{"source":{"id":"1610.07731","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-10-25T04:43:01Z","cross_cats_sorted":[],"title_canon_sha256":"5e1c81ca58563e3ea5b8d3e7223ffd4303114bfa4d56d4b05c14da4a1db15170","abstract_canon_sha256":"8e6dfd4b87063ad6d3834a4d336bdd25a63c3f4bb25496a18b340f0cd8f9878f"},"schema_version":"1.0"},"canonical_sha256":"04a1e02a5d6cb50aa33fd338867965e7db2d4b54f7080ae8b410737d5b495564","source":{"kind":"arxiv","id":"1610.07731","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.07731","created_at":"2026-05-18T00:12:40Z"},{"alias_kind":"arxiv_version","alias_value":"1610.07731v3","created_at":"2026-05-18T00:12:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.07731","created_at":"2026-05-18T00:12:40Z"},{"alias_kind":"pith_short_12","alias_value":"ASQ6AKS5NS2Q","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"ASQ6AKS5NS2QVIZ7","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"ASQ6AKS5","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ASQ6AKS5NS2QVIZ72M4IM6LF47","target":"record","payload":{"canonical_record":{"source":{"id":"1610.07731","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-10-25T04:43:01Z","cross_cats_sorted":[],"title_canon_sha256":"5e1c81ca58563e3ea5b8d3e7223ffd4303114bfa4d56d4b05c14da4a1db15170","abstract_canon_sha256":"8e6dfd4b87063ad6d3834a4d336bdd25a63c3f4bb25496a18b340f0cd8f9878f"},"schema_version":"1.0"},"canonical_sha256":"04a1e02a5d6cb50aa33fd338867965e7db2d4b54f7080ae8b410737d5b495564","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:40.904574Z","signature_b64":"U/hhDcd3kzzKpV8ZwKNytxI6PV8gwcQdR/E8VSg4lnh4tPc0VmL3NjQAcdRFNCiv9Sb23YgRovUJLQ3jCUtqDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04a1e02a5d6cb50aa33fd338867965e7db2d4b54f7080ae8b410737d5b495564","last_reissued_at":"2026-05-18T00:12:40.904096Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:40.904096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.07731","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:12:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1Rccdnswh+L991jIJQsyjOgPAN09ri6I7U5CqfELpY8+yiFDcHv6pdaV+BwLegA8cG9DSfxYrjPms9DH+yqTAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T21:03:27.024571Z"},"content_sha256":"80d9f496a006059c42b6ef65f52d46fd60998cbbcb29a28c1a430d529710558b","schema_version":"1.0","event_id":"sha256:80d9f496a006059c42b6ef65f52d46fd60998cbbcb29a28c1a430d529710558b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ASQ6AKS5NS2QVIZ72M4IM6LF47","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spacious knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Autumn E. Kent, Jessica S. Purcell","submitted_at":"2016-10-25T04:43:01Z","abstract_excerpt":"We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0 on injectivity radius, consider the set of points with injectivity radius at least R; we call this the R-thick part of the manifold. We show that for any $\\epsilon>0$, there exists a knot K in the 3-sphere so that the ratio of the volume of the R-thick part of the knot complement to the volume of the knot complement is at least $1-\\epsilon$. As R approaches"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07731","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:12:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MfjBRY2nuCY+2jlcxOpu4v+961OfkC5umU0Uiq+Pgg/LezMGeu7vFm3PARDm3c55SJ8yj8dHtwfPJCVQzTT8Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T21:03:27.024918Z"},"content_sha256":"fa4b1999e78e35a0c28c2f0b46d3f8a012ce7c1a7593d74cc3c03087af9d78e9","schema_version":"1.0","event_id":"sha256:fa4b1999e78e35a0c28c2f0b46d3f8a012ce7c1a7593d74cc3c03087af9d78e9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ASQ6AKS5NS2QVIZ72M4IM6LF47/bundle.json","state_url":"https://pith.science/pith/ASQ6AKS5NS2QVIZ72M4IM6LF47/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ASQ6AKS5NS2QVIZ72M4IM6LF47/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":"2026-06-03T21:03:27Z","links":{"resolver":"https://pith.science/pith/ASQ6AKS5NS2QVIZ72M4IM6LF47","bundle":"https://pith.science/pith/ASQ6AKS5NS2QVIZ72M4IM6LF47/bundle.json","state":"https://pith.science/pith/ASQ6AKS5NS2QVIZ72M4IM6LF47/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ASQ6AKS5NS2QVIZ72M4IM6LF47/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ASQ6AKS5NS2QVIZ72M4IM6LF47","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":"8e6dfd4b87063ad6d3834a4d336bdd25a63c3f4bb25496a18b340f0cd8f9878f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-10-25T04:43:01Z","title_canon_sha256":"5e1c81ca58563e3ea5b8d3e7223ffd4303114bfa4d56d4b05c14da4a1db15170"},"schema_version":"1.0","source":{"id":"1610.07731","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.07731","created_at":"2026-05-18T00:12:40Z"},{"alias_kind":"arxiv_version","alias_value":"1610.07731v3","created_at":"2026-05-18T00:12:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.07731","created_at":"2026-05-18T00:12:40Z"},{"alias_kind":"pith_short_12","alias_value":"ASQ6AKS5NS2Q","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"ASQ6AKS5NS2QVIZ7","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"ASQ6AKS5","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:fa4b1999e78e35a0c28c2f0b46d3f8a012ce7c1a7593d74cc3c03087af9d78e9","target":"graph","created_at":"2026-05-18T00:12:40Z","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 show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0 on injectivity radius, consider the set of points with injectivity radius at least R; we call this the R-thick part of the manifold. We show that for any $\\epsilon>0$, there exists a knot K in the 3-sphere so that the ratio of the volume of the R-thick part of the knot complement to the volume of the knot complement is at least $1-\\epsilon$. As R approaches","authors_text":"Autumn E. Kent, Jessica S. Purcell","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-10-25T04:43:01Z","title":"Spacious knots"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07731","kind":"arxiv","version":3},"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":"sha256:80d9f496a006059c42b6ef65f52d46fd60998cbbcb29a28c1a430d529710558b","target":"record","created_at":"2026-05-18T00:12:40Z","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":"8e6dfd4b87063ad6d3834a4d336bdd25a63c3f4bb25496a18b340f0cd8f9878f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-10-25T04:43:01Z","title_canon_sha256":"5e1c81ca58563e3ea5b8d3e7223ffd4303114bfa4d56d4b05c14da4a1db15170"},"schema_version":"1.0","source":{"id":"1610.07731","kind":"arxiv","version":3}},"canonical_sha256":"04a1e02a5d6cb50aa33fd338867965e7db2d4b54f7080ae8b410737d5b495564","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"04a1e02a5d6cb50aa33fd338867965e7db2d4b54f7080ae8b410737d5b495564","first_computed_at":"2026-05-18T00:12:40.904096Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:40.904096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U/hhDcd3kzzKpV8ZwKNytxI6PV8gwcQdR/E8VSg4lnh4tPc0VmL3NjQAcdRFNCiv9Sb23YgRovUJLQ3jCUtqDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:40.904574Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.07731","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:80d9f496a006059c42b6ef65f52d46fd60998cbbcb29a28c1a430d529710558b","sha256:fa4b1999e78e35a0c28c2f0b46d3f8a012ce7c1a7593d74cc3c03087af9d78e9"],"state_sha256":"bee92993883b35a5c57b3e543eaee97a7a6b06161cfc77a8660b95585b3ca66a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VgThiS9HD+9cP7XKPUjOdJO5SWmHMUhuwMyIg0Fe6/BYbqkMNyV3D4/j5APO/TDoEr+iWtFPT64P6rQdxiZ5BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T21:03:27.027039Z","bundle_sha256":"3bb6c22be93d05e37212a840210097697879cc378b9d680f8c110a305a9b44dd"}}