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The \\emph{Joker} is the cyclic $\\mathcal{A}(1)^*$-module $\\mathcal{A}(1)^*/\\mathcal{A}(1)^*\\{\\mathrm{Sq}^3\\}$ which plays a special r\\^ole in the study of $\\mathcal{A}(1)^*$-modules. We discuss realisations of the Joker both as an $\\mathcal{A}^*$-module and as the cohomology of a spectrum. We also consider analogous $\\mathcal{A}(n)^*$-modules for $n\\geq2$ and prove realisability results (both stable and unstable) for $n=2,3$ and non-realisability resul"},"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":"1710.02974","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2017-10-09T07:36:52Z","cross_cats_sorted":[],"title_canon_sha256":"d173fea62e941b864c035e4d9a651e83b49c2d486fc202d4e9004508b72214ba","abstract_canon_sha256":"c5133d7bb4788334d39c3905097a129b6ec6a654ce36fefb64880027ebcf55ab"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:50.400350Z","signature_b64":"M/Rm799SvJgsztfvoW0RzZUxUFObNk2zH26pnEpkOQ8OAylx1EQaXGXvvK4GeHXlc8fboJLUBn7aYXlAiVjFCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"477de90c4ce884508fd9717dd699bdb6cb114077f28fcebff6c93d0fd2223825","last_reissued_at":"2026-05-18T00:11:50.399577Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:50.399577Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Iterated doubles of the Joker and their realisability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Andrew Baker","submitted_at":"2017-10-09T07:36:52Z","abstract_excerpt":"Let $\\mathcal{A}(1)^*$ be the subHopf algebra of the mod~$2$ Steenrod algebra $\\mathcal{A}^*$ generated by $\\mathrm{Sq}^1$ and $\\mathrm{Sq}^2$. 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