{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:DA6TLV5MKGC76EXEXPBP2BZ5U2","short_pith_number":"pith:DA6TLV5M","canonical_record":{"source":{"id":"1908.01942","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-06T03:45:06Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"95e5e69afef1c69e1656f32517102b3979abb9454244c161885385436e369374","abstract_canon_sha256":"f0589131e033f077e63bad17f411a8cc7c00859c851c04ad54b60ca6cfefc873"},"schema_version":"1.0"},"canonical_sha256":"183d35d7ac5185ff12e4bbc2fd073da6a820e257c5507e2896c7ee71e378f047","source":{"kind":"arxiv","id":"1908.01942","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.01942","created_at":"2026-07-05T02:28:15Z"},{"alias_kind":"arxiv_version","alias_value":"1908.01942v6","created_at":"2026-07-05T02:28:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01942","created_at":"2026-07-05T02:28:15Z"},{"alias_kind":"pith_short_12","alias_value":"DA6TLV5MKGC7","created_at":"2026-07-05T02:28:15Z"},{"alias_kind":"pith_short_16","alias_value":"DA6TLV5MKGC76EXE","created_at":"2026-07-05T02:28:15Z"},{"alias_kind":"pith_short_8","alias_value":"DA6TLV5M","created_at":"2026-07-05T02:28:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:DA6TLV5MKGC76EXEXPBP2BZ5U2","target":"record","payload":{"canonical_record":{"source":{"id":"1908.01942","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-06T03:45:06Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"95e5e69afef1c69e1656f32517102b3979abb9454244c161885385436e369374","abstract_canon_sha256":"f0589131e033f077e63bad17f411a8cc7c00859c851c04ad54b60ca6cfefc873"},"schema_version":"1.0"},"canonical_sha256":"183d35d7ac5185ff12e4bbc2fd073da6a820e257c5507e2896c7ee71e378f047","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:28:15.515030Z","signature_b64":"f3hEN2oo+pDLbYULM5aWq0pj0JgfJCvZTLsraHRJ/BNUUES9fHog1Bp7//KcjUVMUlTzgdJSENRJe6zAyXAFBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"183d35d7ac5185ff12e4bbc2fd073da6a820e257c5507e2896c7ee71e378f047","last_reissued_at":"2026-07-05T02:28:15.514611Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:28:15.514611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.01942","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:28:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CcMp5OaB1fGWCPLc97SgeoZzVjJha1vsELhsLvaCOxiUls7+Nk6kuCs+xz7/Z1Na37KS1oDKuR6mEpffDRcCCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:33:18.372118Z"},"content_sha256":"4bd2eaf96fd2c40c82c2f4b96115606035787f56fb968d054629448f157bf5bb","schema_version":"1.0","event_id":"sha256:4bd2eaf96fd2c40c82c2f4b96115606035787f56fb968d054629448f157bf5bb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:DA6TLV5MKGC76EXEXPBP2BZ5U2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A lower bound on critical points of the electric potential of a knot","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DS","authors_text":"Max Lipton","submitted_at":"2019-08-06T03:45:06Z","abstract_excerpt":"Given a knot $K$ parametrized by $r: [0,2\\pi] \\to \\mathbb{R}^3$, we can define the electric potential on its complement by $\\Phi(x) = \\int_0^{2\\pi} \\frac{|r'(t)|}{|x - r(t)|}dt$. Physicists and knot theorists want to understand the critical points of the potential and their behavior.\n  The tunneling number $t(K)$ of a knot is the smallest number of arcs one needs to add to a knot so the complement is a handlebody. We show the number of critical points of the potential is at least $2t(K) + 2$. The result is proven using Morse theory and stable manifold theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01942","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.01942/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T02:28:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m5D4H+eNmoAVRSjQVC/SzpZuz21fsQ1Ik1o1yqjQBtPfJW+V11LvBwj37a+B6BAPLZt5eFA+uF9WnVFvRpr8Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:33:18.375925Z"},"content_sha256":"bd2ce5fd49038ebe300564f85b8baa8c2a60ae71001390b4947f2824c82cfaac","schema_version":"1.0","event_id":"sha256:bd2ce5fd49038ebe300564f85b8baa8c2a60ae71001390b4947f2824c82cfaac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/bundle.json","state_url":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/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":"2026-08-16T13:33:18Z","links":{"resolver":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2","bundle":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/bundle.json","state":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:DA6TLV5MKGC76EXEXPBP2BZ5U2","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":"f0589131e033f077e63bad17f411a8cc7c00859c851c04ad54b60ca6cfefc873","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-06T03:45:06Z","title_canon_sha256":"95e5e69afef1c69e1656f32517102b3979abb9454244c161885385436e369374"},"schema_version":"1.0","source":{"id":"1908.01942","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.01942","created_at":"2026-07-05T02:28:15Z"},{"alias_kind":"arxiv_version","alias_value":"1908.01942v6","created_at":"2026-07-05T02:28:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01942","created_at":"2026-07-05T02:28:15Z"},{"alias_kind":"pith_short_12","alias_value":"DA6TLV5MKGC7","created_at":"2026-07-05T02:28:15Z"},{"alias_kind":"pith_short_16","alias_value":"DA6TLV5MKGC76EXE","created_at":"2026-07-05T02:28:15Z"},{"alias_kind":"pith_short_8","alias_value":"DA6TLV5M","created_at":"2026-07-05T02:28:15Z"}],"graph_snapshots":[{"event_id":"sha256:bd2ce5fd49038ebe300564f85b8baa8c2a60ae71001390b4947f2824c82cfaac","target":"graph","created_at":"2026-07-05T02:28:15Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.01942/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a knot $K$ parametrized by $r: [0,2\\pi] \\to \\mathbb{R}^3$, we can define the electric potential on its complement by $\\Phi(x) = \\int_0^{2\\pi} \\frac{|r'(t)|}{|x - r(t)|}dt$. Physicists and knot theorists want to understand the critical points of the potential and their behavior.\n  The tunneling number $t(K)$ of a knot is the smallest number of arcs one needs to add to a knot so the complement is a handlebody. We show the number of critical points of the potential is at least $2t(K) + 2$. The result is proven using Morse theory and stable manifold theory.","authors_text":"Max Lipton","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-06T03:45:06Z","title":"A lower bound on critical points of the electric potential of a knot"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01942","kind":"arxiv","version":6},"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":"sha256:4bd2eaf96fd2c40c82c2f4b96115606035787f56fb968d054629448f157bf5bb","target":"record","created_at":"2026-07-05T02:28:15Z","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":"f0589131e033f077e63bad17f411a8cc7c00859c851c04ad54b60ca6cfefc873","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-06T03:45:06Z","title_canon_sha256":"95e5e69afef1c69e1656f32517102b3979abb9454244c161885385436e369374"},"schema_version":"1.0","source":{"id":"1908.01942","kind":"arxiv","version":6}},"canonical_sha256":"183d35d7ac5185ff12e4bbc2fd073da6a820e257c5507e2896c7ee71e378f047","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"183d35d7ac5185ff12e4bbc2fd073da6a820e257c5507e2896c7ee71e378f047","first_computed_at":"2026-07-05T02:28:15.514611Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:28:15.514611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f3hEN2oo+pDLbYULM5aWq0pj0JgfJCvZTLsraHRJ/BNUUES9fHog1Bp7//KcjUVMUlTzgdJSENRJe6zAyXAFBA==","signature_status":"signed_v1","signed_at":"2026-07-05T02:28:15.515030Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.01942","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4bd2eaf96fd2c40c82c2f4b96115606035787f56fb968d054629448f157bf5bb","sha256:bd2ce5fd49038ebe300564f85b8baa8c2a60ae71001390b4947f2824c82cfaac"],"state_sha256":"c7404fd12cf7b69f3e4d6b929c30ffb25e918f9c8e10cca3646fb4cbec57d61b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c/sjz7WUSB4rM+XhxSgCkHxk2TDL5pfOBJEZ6s35ID430aHqLLKyJ9KTn5DDWqaDJNv3u9q1k3RcCSW13maCDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T13:33:18.409762Z","bundle_sha256":"07c0188459bfe50c9793618db547e3565270c81b8cd7146cec05d9075c5cbf26"}}