{"id":"https://openalex.org/W7204963232","doi":"https://doi.org/10.48550/arxiv.2608.30577","title":"Kempe factorizations for rational curves on $\\operatorname{SO}_4(\\mathbb{R})$","display_name":"Kempe factorizations for rational curves on $\\operatorname{SO}_4(\\mathbb{R})$","publication_year":2026,"publication_date":"2026-08-31","ids":{"openalex":"https://openalex.org/W7204963232","doi":"https://doi.org/10.48550/arxiv.2608.30577"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.30577","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30577","pdf_url":"https://arxiv.org/pdf/2608.30577","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.30577","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5148851330","display_name":"Yifan Li","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Yifan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5001759524","display_name":"Zijia Li","orcid":"https://orcid.org/0000-0002-8734-4396","observed_orcids":["https://orcid.org/0000-0002-8734-4396","https://orcid.org/0009-0008-1354-3465"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Zijia","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5100775242","display_name":"Ke Ye","orcid":"https://orcid.org/0000-0002-3193-7264","observed_orcids":["https://orcid.org/0000-0002-3193-7264"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ye, Ke","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9452999830245972,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9452999830245972,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12037","display_name":"Algebraic and Geometric Analysis","score":0.01720000058412552,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.009700000286102295,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/rational-curves","display_name":"rational curves","score":0.8970000147819519}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8043000102043152},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.58160001039505},{"id":"https://openalex.org/C158158286","wikidata":"https://www.wikidata.org/wiki/Q9009080","display_name":"Invariants of tensors","level":3,"score":0.5202999711036682},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5149999856948853},{"id":"https://openalex.org/C43809302","wikidata":"https://www.wikidata.org/wiki/Q1060874","display_name":"Plane curve","level":2,"score":0.48500001430511475},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4821000099182129},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4625000059604645},{"id":"https://openalex.org/C138029060","wikidata":"https://www.wikidata.org/wiki/Q7226633","display_name":"Polynomial and rational function modeling","level":3,"score":0.42750000953674316},{"id":"https://openalex.org/C200127275","wikidata":"https://www.wikidata.org/wiki/Q173853","display_name":"Quaternion","level":2,"score":0.41659998893737793},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4165000021457672},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4000000059604645},{"id":"https://openalex.org/C156330621","wikidata":"https://www.wikidata.org/wiki/Q3393476","display_name":"Rational point","level":3,"score":0.3889000117778778},{"id":"https://openalex.org/C14103023","wikidata":"https://www.wikidata.org/wiki/Q11681459","display_name":"Pairing","level":3,"score":0.37689998745918274},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.3758000135421753},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3720000088214874},{"id":"https://openalex.org/C75190567","wikidata":"https://www.wikidata.org/wiki/Q41237","display_name":"Rational function","level":2,"score":0.3601999878883362},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.35929998755455017},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.3481999933719635},{"id":"https://openalex.org/C53622353","wikidata":"https://www.wikidata.org/wiki/Q1518913","display_name":"Family of curves","level":2,"score":0.34549999237060547},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3330000042915344},{"id":"https://openalex.org/C179117685","wikidata":"https://www.wikidata.org/wiki/Q328998","display_name":"Complex plane","level":2,"score":0.33160001039505005},{"id":"https://openalex.org/C74050887","wikidata":"https://www.wikidata.org/wiki/Q848368","display_name":"Rotation (mathematics)","level":2,"score":0.3255999982357025},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.3255999982357025},{"id":"https://openalex.org/C70251129","wikidata":"https://www.wikidata.org/wiki/Q260928","display_name":"Jordan curve theorem","level":2,"score":0.31459999084472656},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.31189998984336853},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.3061000108718872},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.28700000047683716},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.2712000012397766},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.263700008392334},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.26190000772476196},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.250900000333786}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.30577","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30577","pdf_url":"https://arxiv.org/pdf/2608.30577","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.30577","doi":"https://doi.org/10.48550/arxiv.2608.30577","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.30577","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.30577","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30577","pdf_url":"https://arxiv.org/pdf/2608.30577","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204963232.pdf","grobid_xml":"https://content.openalex.org/works/W7204963232.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4403966835"],"related_works":["https://openalex.org/W4403966835","https://openalex.org/W4408297961","https://openalex.org/W4297675517","https://openalex.org/W3092530813","https://openalex.org/W4403118568","https://openalex.org/W7147667576","https://openalex.org/W4416436807","https://openalex.org/W7125606269","https://openalex.org/W4388093369","https://openalex.org/W2966458050"],"abstract_inverted_index":{"We":[0],"study":[1],"constructive":[2],"Kempe":[3],"factorizations":[4],"for":[5],"rational":[6,15,26,38],"curves":[7,17,39],"on":[8,18],"$\\operatorname{SO}_4(\\mathbb{R})$.":[9],"Motivated":[10],"by":[11,65,97],"motion-polynomial":[12],"factorization":[13],"and":[14,40,68,81,94],"matrix":[16],"real":[19],"classical":[20],"groups,":[21],"we":[22],"prove":[23],"that":[24],"every":[25],"curve":[27,56],"of":[28,45],"degree":[29],"$2d$":[30,48],"with":[31,85],"$d\\ge1$":[32],"first":[33],"factors":[34,84],"into":[35,42],"$d$":[36],"quadratic":[37],"then":[41],"a":[43,58],"product":[44],"at":[46],"most":[47],"planar":[49,54],"rotation":[50,55],"curves,":[51],"where":[52],"each":[53],"fixes":[57],"two-dimensional":[59],"plane":[60],"pointwise.":[61],"The":[62,89],"construction":[63],"proceeds":[64],"extracting":[66],"left":[67],"right":[69],"isoclinic":[70],"polynomial":[71],"parts":[72],"via":[73],"Cayley's":[74],"factorization,":[75],"factoring":[76],"the":[77],"corresponding":[78],"quaternion":[79],"polynomials,":[80],"pairing":[82],"linear":[83],"equal":[86],"norm":[87],"polynomials.":[88],"resulting":[90],"algorithms":[91],"are":[92,95],"explicit":[93],"illustrated":[96],"examples.":[98]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-02T00:00:00"}