{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:LW3TCE2UORR7ZUIJEVKQGT6RCG","short_pith_number":"pith:LW3TCE2U","canonical_record":{"source":{"id":"1507.07231","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-07-26T18:36:37Z","cross_cats_sorted":[],"title_canon_sha256":"ae5193b7fa19c84e4a0604b7df6a7346745413549034bc4f51c59ee6347c1376","abstract_canon_sha256":"74a30b577a5651be193b02e51337007446f57cd2055307939f63a322d239b6a5"},"schema_version":"1.0"},"canonical_sha256":"5db73113547463fcd1092555034fd111813717cbffccfe21fa88ab5ddee9925c","source":{"kind":"arxiv","id":"1507.07231","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07231","created_at":"2026-05-18T00:54:20Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07231v1","created_at":"2026-05-18T00:54:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07231","created_at":"2026-05-18T00:54:20Z"},{"alias_kind":"pith_short_12","alias_value":"LW3TCE2UORR7","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LW3TCE2UORR7ZUIJ","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LW3TCE2U","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:LW3TCE2UORR7ZUIJEVKQGT6RCG","target":"record","payload":{"canonical_record":{"source":{"id":"1507.07231","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-07-26T18:36:37Z","cross_cats_sorted":[],"title_canon_sha256":"ae5193b7fa19c84e4a0604b7df6a7346745413549034bc4f51c59ee6347c1376","abstract_canon_sha256":"74a30b577a5651be193b02e51337007446f57cd2055307939f63a322d239b6a5"},"schema_version":"1.0"},"canonical_sha256":"5db73113547463fcd1092555034fd111813717cbffccfe21fa88ab5ddee9925c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:20.788986Z","signature_b64":"Ti30XYdPq47uZOzpyFtTiHxYg088ZT2QXTnCQR+weqinGgYZ/x5z82c9rmjUMdKREX7upzoDK2l4sOXWXF4iCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5db73113547463fcd1092555034fd111813717cbffccfe21fa88ab5ddee9925c","last_reissued_at":"2026-05-18T00:54:20.788588Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:20.788588Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.07231","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:54:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+IQFmXyFcnPRJ769Su7MXBzgYi0tmDKo33Zb4lFY1dkeqKhBsBMz3vWOT4JAqBodhJnRHC4QqDxGW4JvyPsUAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:35:48.830755Z"},"content_sha256":"ac92c2607583cbd98b30cb7846162c4b2f955b139acb51f3885398ab710f580d","schema_version":"1.0","event_id":"sha256:ac92c2607583cbd98b30cb7846162c4b2f955b139acb51f3885398ab710f580d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:LW3TCE2UORR7ZUIJEVKQGT6RCG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stabilizing Heegaard Splittings of High-Distance Knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"George Mossessian","submitted_at":"2015-07-26T18:36:37Z","abstract_excerpt":"Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\\choose n}$ distinct minimal genus Heegaard splittings of $S^3\\setminus\\eta(K)$. These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If $K$ has bridge distance at least $4n$, then two splittings from different families become equivalent only after $n-1$ stabilizations. Further, we construct representatives of the isotopy classes of the minimal tunnel systems for $K$ corresponding to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07231","kind":"arxiv","version":1},"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":"2026-05-18T00:54:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t2NUviNh9hewLBjXPpjshdisvMEPzGUKA6sGfKwEFB1KiUmOhMFc522ueferDsQDR2gZrfIPzGGA7sSyc4kLDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:35:48.831097Z"},"content_sha256":"0210932797f92ca4ac6b3966b9759339008866667cff5689a4db8f269f0bb721","schema_version":"1.0","event_id":"sha256:0210932797f92ca4ac6b3966b9759339008866667cff5689a4db8f269f0bb721"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LW3TCE2UORR7ZUIJEVKQGT6RCG/bundle.json","state_url":"https://pith.science/pith/LW3TCE2UORR7ZUIJEVKQGT6RCG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LW3TCE2UORR7ZUIJEVKQGT6RCG/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":"2026-06-09T16:35:48Z","links":{"resolver":"https://pith.science/pith/LW3TCE2UORR7ZUIJEVKQGT6RCG","bundle":"https://pith.science/pith/LW3TCE2UORR7ZUIJEVKQGT6RCG/bundle.json","state":"https://pith.science/pith/LW3TCE2UORR7ZUIJEVKQGT6RCG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LW3TCE2UORR7ZUIJEVKQGT6RCG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:LW3TCE2UORR7ZUIJEVKQGT6RCG","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":"74a30b577a5651be193b02e51337007446f57cd2055307939f63a322d239b6a5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-07-26T18:36:37Z","title_canon_sha256":"ae5193b7fa19c84e4a0604b7df6a7346745413549034bc4f51c59ee6347c1376"},"schema_version":"1.0","source":{"id":"1507.07231","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07231","created_at":"2026-05-18T00:54:20Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07231v1","created_at":"2026-05-18T00:54:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07231","created_at":"2026-05-18T00:54:20Z"},{"alias_kind":"pith_short_12","alias_value":"LW3TCE2UORR7","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LW3TCE2UORR7ZUIJ","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LW3TCE2U","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:0210932797f92ca4ac6b3966b9759339008866667cff5689a4db8f269f0bb721","target":"graph","created_at":"2026-05-18T00:54:20Z","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":"Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\\choose n}$ distinct minimal genus Heegaard splittings of $S^3\\setminus\\eta(K)$. These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If $K$ has bridge distance at least $4n$, then two splittings from different families become equivalent only after $n-1$ stabilizations. Further, we construct representatives of the isotopy classes of the minimal tunnel systems for $K$ corresponding to ","authors_text":"George Mossessian","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-07-26T18:36:37Z","title":"Stabilizing Heegaard Splittings of High-Distance Knots"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07231","kind":"arxiv","version":1},"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:ac92c2607583cbd98b30cb7846162c4b2f955b139acb51f3885398ab710f580d","target":"record","created_at":"2026-05-18T00:54:20Z","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":"74a30b577a5651be193b02e51337007446f57cd2055307939f63a322d239b6a5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-07-26T18:36:37Z","title_canon_sha256":"ae5193b7fa19c84e4a0604b7df6a7346745413549034bc4f51c59ee6347c1376"},"schema_version":"1.0","source":{"id":"1507.07231","kind":"arxiv","version":1}},"canonical_sha256":"5db73113547463fcd1092555034fd111813717cbffccfe21fa88ab5ddee9925c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5db73113547463fcd1092555034fd111813717cbffccfe21fa88ab5ddee9925c","first_computed_at":"2026-05-18T00:54:20.788588Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:54:20.788588Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ti30XYdPq47uZOzpyFtTiHxYg088ZT2QXTnCQR+weqinGgYZ/x5z82c9rmjUMdKREX7upzoDK2l4sOXWXF4iCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:54:20.788986Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.07231","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ac92c2607583cbd98b30cb7846162c4b2f955b139acb51f3885398ab710f580d","sha256:0210932797f92ca4ac6b3966b9759339008866667cff5689a4db8f269f0bb721"],"state_sha256":"14b406766ea3b8e0c1ae0a4bb5eaed9696ba1664517ac325b48ef069635072d9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/STLvxsE8jGxAz5rmf0nfKNJChAk6BpxNSDbblF3cKq3/DPUidWr2LBaL3h06kUoGZ9BaNRmuIgNWG0uSy45AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T16:35:48.833030Z","bundle_sha256":"fabc4d882fc5de7e00467f5658458d4ec457d64c07caa45b6eeaa99e12121104"}}