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We prove that the coefficient of $v^{w-n+1}$ in the Homfly polynomial of the closure of $\\beta$ agrees with $(-1)^{n-1}$ times the coefficient of $v^{w+n^2-1}$ in the Homfly polynomial of the closure of $\\beta\\Delta^2$. This coincidence implies that the lower Morton--Franks-Williams estimate for the $v$--degree of the Homfly polynomial of $\\hat\\beta$ is sharp if and only if the upper MFW estimate is sharp for the $v$--degree of the Homfly polynomial of $\\hat{\\beta\\Delta^2}$."},"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":"0803.0103","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2008-03-02T06:46:46Z","cross_cats_sorted":[],"title_canon_sha256":"ce88234ead7aec9c6784c114e24ec831dfb534ee865b20cb8bda36a3317a741f","abstract_canon_sha256":"fe0bb49c80458636ffc98b63e36b61eea784e198a959cd29eb897c083d55b1e7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:00.681814Z","signature_b64":"QLW6nkcD/gsnkFo+TXrgssnVJBpgfhYlEE8a7QBW8ZsuVDSbvinfJAzwGj7Yd9KxmO9IuMWRifbxbZCoYQb9Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a08504fdb327b33001ccafdfc8c18a97d890f2a5d45e0836d0913600db31ff4a","last_reissued_at":"2026-05-18T02:16:00.681426Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:00.681426Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Meridian twisting of closed braids and the Homfly polynomial","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Tam\\'as K\\'alm\\'an","submitted_at":"2008-03-02T06:46:46Z","abstract_excerpt":"Let $\\beta$ be a braid on $n$ strands, with exponent sum $w$. Let $\\Delta$ be the Garside half-twist braid. We prove that the coefficient of $v^{w-n+1}$ in the Homfly polynomial of the closure of $\\beta$ agrees with $(-1)^{n-1}$ times the coefficient of $v^{w+n^2-1}$ in the Homfly polynomial of the closure of $\\beta\\Delta^2$. This coincidence implies that the lower Morton--Franks-Williams estimate for the $v$--degree of the Homfly polynomial of $\\hat\\beta$ is sharp if and only if the upper MFW estimate is sharp for the $v$--degree of the Homfly polynomial of $\\hat{\\beta\\Delta^2}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0803.0103","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":"0803.0103","created_at":"2026-05-18T02:16:00.681486+00:00"},{"alias_kind":"arxiv_version","alias_value":"0803.0103v1","created_at":"2026-05-18T02:16:00.681486+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0803.0103","created_at":"2026-05-18T02:16:00.681486+00:00"},{"alias_kind":"pith_short_12","alias_value":"UCCQJ7NTE6ZT","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_16","alias_value":"UCCQJ7NTE6ZTAAOM","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_8","alias_value":"UCCQJ7NT","created_at":"2026-05-18T12:25:58.018023+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/UCCQJ7NTE6ZTAAOMV7P4RQMKS7","json":"https://pith.science/pith/UCCQJ7NTE6ZTAAOMV7P4RQMKS7.json","graph_json":"https://pith.science/api/pith-number/UCCQJ7NTE6ZTAAOMV7P4RQMKS7/graph.json","events_json":"https://pith.science/api/pith-number/UCCQJ7NTE6ZTAAOMV7P4RQMKS7/events.json","paper":"https://pith.science/paper/UCCQJ7NT"},"agent_actions":{"view_html":"https://pith.science/pith/UCCQJ7NTE6ZTAAOMV7P4RQMKS7","download_json":"https://pith.science/pith/UCCQJ7NTE6ZTAAOMV7P4RQMKS7.json","view_paper":"https://pith.science/paper/UCCQJ7NT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0803.0103&json=true","fetch_graph":"https://pith.science/api/pith-number/UCCQJ7NTE6ZTAAOMV7P4RQMKS7/graph.json","fetch_events":"https://pith.science/api/pith-number/UCCQJ7NTE6ZTAAOMV7P4RQMKS7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UCCQJ7NTE6ZTAAOMV7P4RQMKS7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UCCQJ7NTE6ZTAAOMV7P4RQMKS7/action/storage_attestation","attest_author":"https://pith.science/pith/UCCQJ7NTE6ZTAAOMV7P4RQMKS7/action/author_attestation","sign_citation":"https://pith.science/pith/UCCQJ7NTE6ZTAAOMV7P4RQMKS7/action/citation_signature","submit_replication":"https://pith.science/pith/UCCQJ7NTE6ZTAAOMV7P4RQMKS7/action/replication_record"}},"created_at":"2026-05-18T02:16:00.681486+00:00","updated_at":"2026-05-18T02:16:00.681486+00:00"}