{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:DA6TLV5MKGC76EXEXPBP2BZ5U2","short_pith_number":"pith:DA6TLV5M","schema_version":"1.0","canonical_sha256":"183d35d7ac5185ff12e4bbc2fd073da6a820e257c5507e2896c7ee71e378f047","source":{"kind":"arxiv","id":"1908.01942","version":6},"attestation_state":"computed","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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.01942","created_at":"2026-07-05T02:28:15.514671+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.01942v6","created_at":"2026-07-05T02:28:15.514671+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01942","created_at":"2026-07-05T02:28:15.514671+00:00"},{"alias_kind":"pith_short_12","alias_value":"DA6TLV5MKGC7","created_at":"2026-07-05T02:28:15.514671+00:00"},{"alias_kind":"pith_short_16","alias_value":"DA6TLV5MKGC76EXE","created_at":"2026-07-05T02:28:15.514671+00:00"},{"alias_kind":"pith_short_8","alias_value":"DA6TLV5M","created_at":"2026-07-05T02:28:15.514671+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2","json":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2.json","graph_json":"https://pith.science/api/pith-number/DA6TLV5MKGC76EXEXPBP2BZ5U2/graph.json","events_json":"https://pith.science/api/pith-number/DA6TLV5MKGC76EXEXPBP2BZ5U2/events.json","paper":"https://pith.science/paper/DA6TLV5M"},"agent_actions":{"view_html":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2","download_json":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2.json","view_paper":"https://pith.science/paper/DA6TLV5M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.01942&json=true","fetch_graph":"https://pith.science/api/pith-number/DA6TLV5MKGC76EXEXPBP2BZ5U2/graph.json","fetch_events":"https://pith.science/api/pith-number/DA6TLV5MKGC76EXEXPBP2BZ5U2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/action/storage_attestation","attest_author":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/action/author_attestation","sign_citation":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/action/citation_signature","submit_replication":"https://pith.science/pith/DA6TLV5MKGC76EXEXPBP2BZ5U2/action/replication_record"}},"created_at":"2026-07-05T02:28:15.514671+00:00","updated_at":"2026-07-05T02:28:15.514671+00:00"}