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Moreover, let $\\mathcal{D}=\\{0,v,2v,\\ldots,(B-1)v\\}$ for some $v\\in\\mathbb{Z}^2$ such that $v,M v$ are linearly independent. We are interested in the topological properties of the self-affine tile $\\mathcal{T}$ defined by $M\\mathcal{T}=\\bigcup_{d\\in\\mathcal{D}}(\\mathcal{T}+d)$. Lau and Leung proved that $\\mathcal{T}$ is homeomorphic to a closed disk if and only if $2|A|\\le"},"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":"1801.02957","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-01-09T14:43:41Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"e3785c15d7bd8f8cf555acfe17b73ca82b08734250fd295d29291e21452b9425","abstract_canon_sha256":"6db9e4bf18bdbdbec817092f299b15582d537c4620d4b01e9d05001dde43ede8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:21.921415Z","signature_b64":"SyKzhwC4L1/O5p+CkbFlLvY97OvQNdPSRBoNWftRR8DRrzuIb8nwVZ8L6RPeBGuAlYtvevKvSnwpF06z126UDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b96ef6a288aa11bc4cf619263fbb87fb671563801c900df23d6281d9e978ed4b","last_reissued_at":"2026-05-18T00:26:21.920801Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:21.920801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topology of planar self-affine tiles with collinear digit set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.GN","authors_text":"Beno\\^it Loridant, J\\\"org Thuswaldner, Shigeki Akiyama","submitted_at":"2018-01-09T14:43:41Z","abstract_excerpt":"We consider the self-affine tiles with collinear digit set defined as follows. Let $A,B\\in\\mathbb{Z}$ satisfy $|A|\\leq B\\geq 2$ and $M\\in\\mathbb{Z}^{2\\times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $\\mathcal{D}=\\{0,v,2v,\\ldots,(B-1)v\\}$ for some $v\\in\\mathbb{Z}^2$ such that $v,M v$ are linearly independent. We are interested in the topological properties of the self-affine tile $\\mathcal{T}$ defined by $M\\mathcal{T}=\\bigcup_{d\\in\\mathcal{D}}(\\mathcal{T}+d)$. Lau and Leung proved that $\\mathcal{T}$ is homeomorphic to a closed disk if and only if $2|A|\\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02957","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":"1801.02957","created_at":"2026-05-18T00:26:21.920885+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.02957v1","created_at":"2026-05-18T00:26:21.920885+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.02957","created_at":"2026-05-18T00:26:21.920885+00:00"},{"alias_kind":"pith_short_12","alias_value":"XFXPNIUIVII3","created_at":"2026-05-18T12:33:01.666342+00:00"},{"alias_kind":"pith_short_16","alias_value":"XFXPNIUIVII3YTHW","created_at":"2026-05-18T12:33:01.666342+00:00"},{"alias_kind":"pith_short_8","alias_value":"XFXPNIUI","created_at":"2026-05-18T12:33:01.666342+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/XFXPNIUIVII3YTHWDETD7O4H7N","json":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N.json","graph_json":"https://pith.science/api/pith-number/XFXPNIUIVII3YTHWDETD7O4H7N/graph.json","events_json":"https://pith.science/api/pith-number/XFXPNIUIVII3YTHWDETD7O4H7N/events.json","paper":"https://pith.science/paper/XFXPNIUI"},"agent_actions":{"view_html":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N","download_json":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N.json","view_paper":"https://pith.science/paper/XFXPNIUI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.02957&json=true","fetch_graph":"https://pith.science/api/pith-number/XFXPNIUIVII3YTHWDETD7O4H7N/graph.json","fetch_events":"https://pith.science/api/pith-number/XFXPNIUIVII3YTHWDETD7O4H7N/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N/action/storage_attestation","attest_author":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N/action/author_attestation","sign_citation":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N/action/citation_signature","submit_replication":"https://pith.science/pith/XFXPNIUIVII3YTHWDETD7O4H7N/action/replication_record"}},"created_at":"2026-05-18T00:26:21.920885+00:00","updated_at":"2026-05-18T00:26:21.920885+00:00"}