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We prove that $\\widehat{HF}_{\\mathbb{Z}_{2}}(\\Sigma(K))$ can be computed from knot Heegaard diagrams of $K$ and show that it is a strong Heegaard invariant. As a topolocial application, we construct a transverse knot invariant $\\hat{\\mathcal{T}}_{\\mathbb{Z}_{2}}(K)$ as an element of $\\widehat{HFK}_{\\mathbb{Z}_{2}}(\\Sigma(K),K)$, which"},"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":"1810.01919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-10-03T19:04:02Z","cross_cats_sorted":[],"title_canon_sha256":"ecb404ee915f973a71daf5b7f48acf1ceeefd858c3963baf10c254b217155dba","abstract_canon_sha256":"43b4f7d5848adc592d3e465b5b54aeea9148fa7eff613d1b2c1a35d050392e94"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:06.968962Z","signature_b64":"9FQ3Khu3z2z1X35sAL9HRZqsNLMldfpXC09xpvZxYmr7PvuW83ruE8IfeS8URpGkqyOzxet/8KKTvPDuVdssBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c44ef13f6a69d7b67d34219db1b729a5337f4bee96ecc5e6f141807f00bedba","last_reissued_at":"2026-05-18T00:04:06.968143Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:06.968143Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\mathbb{Z}_{2}$-equivariant Heegaard Floer cohomology of knots in $S^{3}$ as a strong Heegaard invariant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Sungkyung Kang","submitted_at":"2018-10-03T19:04:02Z","abstract_excerpt":"The $\\mathbb{Z}_{2}$-equivariant Heegaard Floer cohomlogy $\\widehat{HF}_{\\mathbb{Z}_{2}}(\\Sigma(K))$ of a knot $K$ in $S^{3}$, constructed by Hendricks, Lipshitz, and Sarkar, is an isotopy invariant which is defined using bridge diagrams of $K$ drawn on a sphere. We prove that $\\widehat{HF}_{\\mathbb{Z}_{2}}(\\Sigma(K))$ can be computed from knot Heegaard diagrams of $K$ and show that it is a strong Heegaard invariant. As a topolocial application, we construct a transverse knot invariant $\\hat{\\mathcal{T}}_{\\mathbb{Z}_{2}}(K)$ as an element of $\\widehat{HFK}_{\\mathbb{Z}_{2}}(\\Sigma(K),K)$, which"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.01919","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1810.01919","created_at":"2026-05-18T00:04:06.968276+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.01919v1","created_at":"2026-05-18T00:04:06.968276+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.01919","created_at":"2026-05-18T00:04:06.968276+00:00"},{"alias_kind":"pith_short_12","alias_value":"JRCO6E7WU2OX","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_16","alias_value":"JRCO6E7WU2OXWZ6T","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_8","alias_value":"JRCO6E7W","created_at":"2026-05-18T12:32:31.084164+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/JRCO6E7WU2OXWZ6TIIM5WG3STJ","json":"https://pith.science/pith/JRCO6E7WU2OXWZ6TIIM5WG3STJ.json","graph_json":"https://pith.science/api/pith-number/JRCO6E7WU2OXWZ6TIIM5WG3STJ/graph.json","events_json":"https://pith.science/api/pith-number/JRCO6E7WU2OXWZ6TIIM5WG3STJ/events.json","paper":"https://pith.science/paper/JRCO6E7W"},"agent_actions":{"view_html":"https://pith.science/pith/JRCO6E7WU2OXWZ6TIIM5WG3STJ","download_json":"https://pith.science/pith/JRCO6E7WU2OXWZ6TIIM5WG3STJ.json","view_paper":"https://pith.science/paper/JRCO6E7W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.01919&json=true","fetch_graph":"https://pith.science/api/pith-number/JRCO6E7WU2OXWZ6TIIM5WG3STJ/graph.json","fetch_events":"https://pith.science/api/pith-number/JRCO6E7WU2OXWZ6TIIM5WG3STJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JRCO6E7WU2OXWZ6TIIM5WG3STJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JRCO6E7WU2OXWZ6TIIM5WG3STJ/action/storage_attestation","attest_author":"https://pith.science/pith/JRCO6E7WU2OXWZ6TIIM5WG3STJ/action/author_attestation","sign_citation":"https://pith.science/pith/JRCO6E7WU2OXWZ6TIIM5WG3STJ/action/citation_signature","submit_replication":"https://pith.science/pith/JRCO6E7WU2OXWZ6TIIM5WG3STJ/action/replication_record"}},"created_at":"2026-05-18T00:04:06.968276+00:00","updated_at":"2026-05-18T00:04:06.968276+00:00"}