{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:L6BAXBERKJFPSYOIBBM3FQNERY","short_pith_number":"pith:L6BAXBER","canonical_record":{"source":{"id":"2606.11301","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-06-09T18:00:03Z","cross_cats_sorted":[],"title_canon_sha256":"ba14b7221135363c5eaff4712919244f068b3d2a7506666b4fad41a85c7b3843","abstract_canon_sha256":"85ca4d733737d604f1c56fda87c0ea9f41954ee1c7ca602a6d56630ceb099bdd"},"schema_version":"1.0"},"canonical_sha256":"5f820b8491524af961c80859b2c1a48e32e7b04c675dfb5ce058e0deffaebda6","source":{"kind":"arxiv","id":"2606.11301","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11301","created_at":"2026-06-11T00:08:17Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11301v1","created_at":"2026-06-11T00:08:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11301","created_at":"2026-06-11T00:08:17Z"},{"alias_kind":"pith_short_12","alias_value":"L6BAXBERKJFP","created_at":"2026-06-11T00:08:17Z"},{"alias_kind":"pith_short_16","alias_value":"L6BAXBERKJFPSYOI","created_at":"2026-06-11T00:08:17Z"},{"alias_kind":"pith_short_8","alias_value":"L6BAXBER","created_at":"2026-06-11T00:08:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:L6BAXBERKJFPSYOIBBM3FQNERY","target":"record","payload":{"canonical_record":{"source":{"id":"2606.11301","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-06-09T18:00:03Z","cross_cats_sorted":[],"title_canon_sha256":"ba14b7221135363c5eaff4712919244f068b3d2a7506666b4fad41a85c7b3843","abstract_canon_sha256":"85ca4d733737d604f1c56fda87c0ea9f41954ee1c7ca602a6d56630ceb099bdd"},"schema_version":"1.0"},"canonical_sha256":"5f820b8491524af961c80859b2c1a48e32e7b04c675dfb5ce058e0deffaebda6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T00:08:17.570352Z","signature_b64":"dVuhgTW2G2YGsxKfFYze72q3Au0Umm4XVRE0Je+reIWD/yxRu1m7yrVUsvY/7sJmYJI2xJYN392jcnNAkZs8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f820b8491524af961c80859b2c1a48e32e7b04c675dfb5ce058e0deffaebda6","last_reissued_at":"2026-06-11T00:08:17.569507Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T00:08:17.569507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.11301","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-11T00:08:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BULMfb3A8RFNXkwovqnFd5O9tWw8t0YwliXn83xjBN80RR12iePMSBHVfXDSxdIa2c5wa4mC8sZLR93BXaquDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T04:28:33.344113Z"},"content_sha256":"a5fd3f8f593253930f9406f44c76f9ee04ff253585167b728b0568e7f3c80781","schema_version":"1.0","event_id":"sha256:a5fd3f8f593253930f9406f44c76f9ee04ff253585167b728b0568e7f3c80781"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:L6BAXBERKJFPSYOIBBM3FQNERY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral Factorization and Hypergeometric Representations of the Alexander Polynomials of $Th(4,2n+1)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Suman Saurabh","submitted_at":"2026-06-09T18:00:03Z","abstract_excerpt":"We study the Alexander polynomials of the 4-strand Turk's head knots $Th(4,2n+1)$, defined as the closures of the braid $(\\sigma_1\\sigma_2^{-1}\\sigma_3)^{2n+1}$. Using the reduced Burau representation, we derive an annihilating recurrence of order at most 8 and a rational generating function for the resulting polynomial sequence. By executing a multivariable resultant elimination over the reciprocal constraint, we obtain an exact factorization of the normalized Alexander polynomial in terms of Chebyshev polynomials. This factorization produces a binomial convolution formula for an associated c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11301","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.11301/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":"2026-06-11T00:08:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tIVNhvc38VU7nzLF8w7C6jS9rWZZhTdSZUL24z+eOLp9+VAWeOYfW5I5gyvISBkwW8ebh+VN+3NbI89GIrvnDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T04:28:33.344517Z"},"content_sha256":"2458ad3f52d63113fd5dad744e611657c36247b54fbdf753d999d95f02cb3ec8","schema_version":"1.0","event_id":"sha256:2458ad3f52d63113fd5dad744e611657c36247b54fbdf753d999d95f02cb3ec8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L6BAXBERKJFPSYOIBBM3FQNERY/bundle.json","state_url":"https://pith.science/pith/L6BAXBERKJFPSYOIBBM3FQNERY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L6BAXBERKJFPSYOIBBM3FQNERY/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":"2026-08-17T04:28:33Z","links":{"resolver":"https://pith.science/pith/L6BAXBERKJFPSYOIBBM3FQNERY","bundle":"https://pith.science/pith/L6BAXBERKJFPSYOIBBM3FQNERY/bundle.json","state":"https://pith.science/pith/L6BAXBERKJFPSYOIBBM3FQNERY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L6BAXBERKJFPSYOIBBM3FQNERY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:L6BAXBERKJFPSYOIBBM3FQNERY","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":"85ca4d733737d604f1c56fda87c0ea9f41954ee1c7ca602a6d56630ceb099bdd","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-06-09T18:00:03Z","title_canon_sha256":"ba14b7221135363c5eaff4712919244f068b3d2a7506666b4fad41a85c7b3843"},"schema_version":"1.0","source":{"id":"2606.11301","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11301","created_at":"2026-06-11T00:08:17Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11301v1","created_at":"2026-06-11T00:08:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11301","created_at":"2026-06-11T00:08:17Z"},{"alias_kind":"pith_short_12","alias_value":"L6BAXBERKJFP","created_at":"2026-06-11T00:08:17Z"},{"alias_kind":"pith_short_16","alias_value":"L6BAXBERKJFPSYOI","created_at":"2026-06-11T00:08:17Z"},{"alias_kind":"pith_short_8","alias_value":"L6BAXBER","created_at":"2026-06-11T00:08:17Z"}],"graph_snapshots":[{"event_id":"sha256:2458ad3f52d63113fd5dad744e611657c36247b54fbdf753d999d95f02cb3ec8","target":"graph","created_at":"2026-06-11T00:08:17Z","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/2606.11301/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the Alexander polynomials of the 4-strand Turk's head knots $Th(4,2n+1)$, defined as the closures of the braid $(\\sigma_1\\sigma_2^{-1}\\sigma_3)^{2n+1}$. Using the reduced Burau representation, we derive an annihilating recurrence of order at most 8 and a rational generating function for the resulting polynomial sequence. By executing a multivariable resultant elimination over the reciprocal constraint, we obtain an exact factorization of the normalized Alexander polynomial in terms of Chebyshev polynomials. This factorization produces a binomial convolution formula for an associated c","authors_text":"Suman Saurabh","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-06-09T18:00:03Z","title":"Spectral Factorization and Hypergeometric Representations of the Alexander Polynomials of $Th(4,2n+1)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11301","kind":"arxiv","version":1},"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:a5fd3f8f593253930f9406f44c76f9ee04ff253585167b728b0568e7f3c80781","target":"record","created_at":"2026-06-11T00:08:17Z","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":"85ca4d733737d604f1c56fda87c0ea9f41954ee1c7ca602a6d56630ceb099bdd","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-06-09T18:00:03Z","title_canon_sha256":"ba14b7221135363c5eaff4712919244f068b3d2a7506666b4fad41a85c7b3843"},"schema_version":"1.0","source":{"id":"2606.11301","kind":"arxiv","version":1}},"canonical_sha256":"5f820b8491524af961c80859b2c1a48e32e7b04c675dfb5ce058e0deffaebda6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f820b8491524af961c80859b2c1a48e32e7b04c675dfb5ce058e0deffaebda6","first_computed_at":"2026-06-11T00:08:17.569507Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T00:08:17.569507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dVuhgTW2G2YGsxKfFYze72q3Au0Umm4XVRE0Je+reIWD/yxRu1m7yrVUsvY/7sJmYJI2xJYN392jcnNAkZs8Cw==","signature_status":"signed_v1","signed_at":"2026-06-11T00:08:17.570352Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11301","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a5fd3f8f593253930f9406f44c76f9ee04ff253585167b728b0568e7f3c80781","sha256:2458ad3f52d63113fd5dad744e611657c36247b54fbdf753d999d95f02cb3ec8"],"state_sha256":"b8f1e33cbed8492b3d930dbe71ab7a396c2953cf92dad15a0be989c0e3d8c488"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UXuS5PuEu/Qg6oup8Qqu8aaE9e5f6AHaVnrkIPXSmwNByOrOT8imRZCV8TS3xFZU77PUBCxtONFVcoBIp82nDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T04:28:33.347066Z","bundle_sha256":"d06a7a67af7e0fbf04475ff46e9a965a96eb543c2dd07a704db2f44657700316"}}