{"paper":{"title":"Stratification of $\\mathrm{AGL}_r(\\mathbb{C})$-representation varieties of twisted Hopf links","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"AGL_r(C) representation varieties of twisted Hopf link complements can be stratified using corresponding GL_r(C) varieties.","cross_cats":[],"primary_cat":"math.GT","authors_text":"\\'Angel Molina-Navarro","submitted_at":"2026-05-13T14:21:34Z","abstract_excerpt":"We provide a stratification of the $\\mathrm{AGL}_r(\\mathbb{C})$-representation variety of the fundamental group of the complement of a twisted Hopf link in terms of a stratification of the corresponding $\\mathrm{GL}_r(\\mathbb{C})$-representation variety. For ranks $1$ and $2$, we explicitly describe this stratification and compute the motives of these varieties in terms of the Lefschetz motive $q=[\\mathbb{C}]$ in the Grothendieck ring of complex algebraic varieties $K_0(\\mathbf{Var}_{\\mathbb{C}})$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We provide a stratification of the AGL_r(C)-representation variety of the fundamental group of the complement of a twisted Hopf link in terms of a stratification of the corresponding GL_r(C)-representation variety. For ranks 1 and 2, we explicitly describe this stratification and compute the motives of these varieties in terms of the Lefschetz motive q=[C] in the Grothendieck ring of complex algebraic varieties K_0(Var_C).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that the AGL_r(C)-representation variety can be stratified directly in terms of the GL_r(C) one, which relies on the specific structure of the fundamental group of the twisted Hopf link complement and properties of algebraic group representations.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The AGL_r(C) representation varieties for twisted Hopf links are stratified using GL_r(C) varieties, with explicit descriptions and motives computed for ranks 1 and 2 in the Grothendieck ring.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"AGL_r(C) representation varieties of twisted Hopf link complements can be stratified using corresponding GL_r(C) varieties.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"55e7e23ae7a605c3a82f0eb37ef914f3012c63064763a3517d544856b692ed0a"},"source":{"id":"2605.13585","kind":"arxiv","version":1},"verdict":{"id":"9e034f59-9eb8-4e51-a11f-0564bd493786","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:57:19.752862Z","strongest_claim":"We provide a stratification of the AGL_r(C)-representation variety of the fundamental group of the complement of a twisted Hopf link in terms of a stratification of the corresponding GL_r(C)-representation variety. For ranks 1 and 2, we explicitly describe this stratification and compute the motives of these varieties in terms of the Lefschetz motive q=[C] in the Grothendieck ring of complex algebraic varieties K_0(Var_C).","one_line_summary":"The AGL_r(C) representation varieties for twisted Hopf links are stratified using GL_r(C) varieties, with explicit descriptions and motives computed for ranks 1 and 2 in the Grothendieck ring.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that the AGL_r(C)-representation variety can be stratified directly in terms of the GL_r(C) one, which relies on the specific structure of the fundamental group of the twisted Hopf link complement and properties of algebraic group representations.","pith_extraction_headline":"AGL_r(C) representation varieties of twisted Hopf link complements can be stratified using corresponding GL_r(C) varieties."},"references":{"count":32,"sample":[{"doi":"","year":1967,"title":"Burde,Darstellungen von Knotengruppen.Mathematische Annalen173(1967), 24-33","work_id":"8d6acc2f-6acf-4c29-bac5-b9ba96cc85b9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Calleja,Configuration spaces of orbits and their Sn-equivariant E-polynomials.arXiv preprint: arXiv:2403.07765v2, 2024","work_id":"ec18a26e-d190-4541-8c36-51d164dba1f7","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"D. Cooper, M. Culler , H. Gillet, D. D. Long, and P. B. Shalen,Plane curves associated to character varieties of3-manifolds.Inventiones mathematicae118(1994), 47-84","work_id":"f6b65dba-a149-4133-9970-8b3642ea9cc4","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"P. R. Cromwell,Knots and Links, Cambridge University Press, 2004","work_id":"27e26262-898c-4a39-8d75-08b2557ace2a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1983,"title":"M. Culler and P. B. Shalen,Varieties of group representations and splitting of3-manifolds.Annals of Mathematics117(1983), n. 1, 109-146","work_id":"cc9fa525-4f65-463d-bce5-e4bc61a98eef","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":32,"snapshot_sha256":"cf68d09e5f502c11083c78803f88d4324b696b4df8da399cd787354132691a99","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"87bb90ea47086f01ef39368a2ac7710d330240d1628a9d7635fa62117d7d9b39"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}