{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:IWLCMTFW4E2MWARMNATAZENIUO","short_pith_number":"pith:IWLCMTFW","schema_version":"1.0","canonical_sha256":"4596264cb6e134cb022c68260c91a8a3be7c5df8633ae9380d6408fc3cbf08bf","source":{"kind":"arxiv","id":"math/0612716","version":2},"attestation_state":"computed","paper":{"title":"The Burau estimate for the entropy of a braid","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DS","authors_text":"Gavin Band, Philip Boyland","submitted_at":"2006-12-22T17:32:31Z","abstract_excerpt":"The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius of the braid's Burau matrix, $B(t)$, after substituting a complex number of modulus~1 in place of $t$. In this paper we show that for a pseudo-Anosov braid the estimate is sharp for the substitution of a"},"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":"math/0612716","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2006-12-22T17:32:31Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"69a552cba677b1b73bf5fe789ce20b4f677228d72b7460e1f441e40bc209b211","abstract_canon_sha256":"437d649ec67ea2dafd7f5dd5ca2bfdc13527b360be2d7230efe170fe1910a110"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:26.645646Z","signature_b64":"Q79C+ETavBV0t1BfcNkMB65pvRjpZvrBhvtIlEEGdU5fhUbZJ4Gzsub+V/yZhegffwGAVMdG791FQW0gkks3Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4596264cb6e134cb022c68260c91a8a3be7c5df8633ae9380d6408fc3cbf08bf","last_reissued_at":"2026-05-18T02:41:26.645127Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:26.645127Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Burau estimate for the entropy of a braid","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DS","authors_text":"Gavin Band, Philip Boyland","submitted_at":"2006-12-22T17:32:31Z","abstract_excerpt":"The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius of the braid's Burau matrix, $B(t)$, after substituting a complex number of modulus~1 in place of $t$. In this paper we show that for a pseudo-Anosov braid the estimate is sharp for the substitution of a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612716","kind":"arxiv","version":2},"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":"math/0612716","created_at":"2026-05-18T02:41:26.645195+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0612716v2","created_at":"2026-05-18T02:41:26.645195+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0612716","created_at":"2026-05-18T02:41:26.645195+00:00"},{"alias_kind":"pith_short_12","alias_value":"IWLCMTFW4E2M","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"IWLCMTFW4E2MWARM","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"IWLCMTFW","created_at":"2026-05-18T12:25:54.717736+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/IWLCMTFW4E2MWARMNATAZENIUO","json":"https://pith.science/pith/IWLCMTFW4E2MWARMNATAZENIUO.json","graph_json":"https://pith.science/api/pith-number/IWLCMTFW4E2MWARMNATAZENIUO/graph.json","events_json":"https://pith.science/api/pith-number/IWLCMTFW4E2MWARMNATAZENIUO/events.json","paper":"https://pith.science/paper/IWLCMTFW"},"agent_actions":{"view_html":"https://pith.science/pith/IWLCMTFW4E2MWARMNATAZENIUO","download_json":"https://pith.science/pith/IWLCMTFW4E2MWARMNATAZENIUO.json","view_paper":"https://pith.science/paper/IWLCMTFW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0612716&json=true","fetch_graph":"https://pith.science/api/pith-number/IWLCMTFW4E2MWARMNATAZENIUO/graph.json","fetch_events":"https://pith.science/api/pith-number/IWLCMTFW4E2MWARMNATAZENIUO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IWLCMTFW4E2MWARMNATAZENIUO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IWLCMTFW4E2MWARMNATAZENIUO/action/storage_attestation","attest_author":"https://pith.science/pith/IWLCMTFW4E2MWARMNATAZENIUO/action/author_attestation","sign_citation":"https://pith.science/pith/IWLCMTFW4E2MWARMNATAZENIUO/action/citation_signature","submit_replication":"https://pith.science/pith/IWLCMTFW4E2MWARMNATAZENIUO/action/replication_record"}},"created_at":"2026-05-18T02:41:26.645195+00:00","updated_at":"2026-05-18T02:41:26.645195+00:00"}