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A knot $n$-mosaic is an $n \\times n$ matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. $D_n$ denotes the total number of all knot $n$-mosaics. Already known is that $D_1=1$, $D_2=2$, and $D_3=22$. In this paper we establish the lower and upper bounds on $D_n$ $$\\frac{2}{275}(9 \\cdot 6^{n-2} + 1)^2 \\cdot 2^{(n-3)^2} \\ \\leq \\"},"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":"1303.7044","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-03-28T06:02:21Z","cross_cats_sorted":[],"title_canon_sha256":"798167f216686b332ed5783ad007b5f6b407f16ef5ca1278efb7c3108fb5d61b","abstract_canon_sha256":"9a238388493907fe5e7dced735423b88bfd14d64c25aec51e62771d6136f8ce7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:17.284991Z","signature_b64":"SWocaJjSu/dwnLoLMJOfIf1ghQCmioLtmCQapmptaD96xXHjMjxm7DFi4cNw1Scbuj7sa/hexFUCmcAxAD3hBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5caefb29599356955d185864e5b0675b0810609c2811056c159f13f5fa7ced1","last_reissued_at":"2026-05-18T02:38:17.284344Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:17.284344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper bound on the total number of knot $n$-mosaics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Ho Lee, Hwa Jeong Lee, Kyungpyo Hong, Seungsang Oh","submitted_at":"2013-03-28T06:02:21Z","abstract_excerpt":"Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. 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In this paper we establish the lower and upper bounds on $D_n$ $$\\frac{2}{275}(9 \\cdot 6^{n-2} + 1)^2 \\cdot 2^{(n-3)^2} \\ \\leq \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.7044","kind":"arxiv","version":3},"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":"1303.7044","created_at":"2026-05-18T02:38:17.284445+00:00"},{"alias_kind":"arxiv_version","alias_value":"1303.7044v3","created_at":"2026-05-18T02:38:17.284445+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.7044","created_at":"2026-05-18T02:38:17.284445+00:00"},{"alias_kind":"pith_short_12","alias_value":"2XFO7MUVTE2W","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_16","alias_value":"2XFO7MUVTE2WSVOR","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_8","alias_value":"2XFO7MUV","created_at":"2026-05-18T12:27:32.513160+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/2XFO7MUVTE2WSVORQWDE4WYGOW","json":"https://pith.science/pith/2XFO7MUVTE2WSVORQWDE4WYGOW.json","graph_json":"https://pith.science/api/pith-number/2XFO7MUVTE2WSVORQWDE4WYGOW/graph.json","events_json":"https://pith.science/api/pith-number/2XFO7MUVTE2WSVORQWDE4WYGOW/events.json","paper":"https://pith.science/paper/2XFO7MUV"},"agent_actions":{"view_html":"https://pith.science/pith/2XFO7MUVTE2WSVORQWDE4WYGOW","download_json":"https://pith.science/pith/2XFO7MUVTE2WSVORQWDE4WYGOW.json","view_paper":"https://pith.science/paper/2XFO7MUV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1303.7044&json=true","fetch_graph":"https://pith.science/api/pith-number/2XFO7MUVTE2WSVORQWDE4WYGOW/graph.json","fetch_events":"https://pith.science/api/pith-number/2XFO7MUVTE2WSVORQWDE4WYGOW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2XFO7MUVTE2WSVORQWDE4WYGOW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2XFO7MUVTE2WSVORQWDE4WYGOW/action/storage_attestation","attest_author":"https://pith.science/pith/2XFO7MUVTE2WSVORQWDE4WYGOW/action/author_attestation","sign_citation":"https://pith.science/pith/2XFO7MUVTE2WSVORQWDE4WYGOW/action/citation_signature","submit_replication":"https://pith.science/pith/2XFO7MUVTE2WSVORQWDE4WYGOW/action/replication_record"}},"created_at":"2026-05-18T02:38:17.284445+00:00","updated_at":"2026-05-18T02:38:17.284445+00:00"}