{"id":"https://openalex.org/W7223676583","doi":"https://doi.org/10.48550/arxiv.2610.32320","title":"The elastic figure-eight knot is not planar","display_name":"The elastic figure-eight knot is not planar","publication_year":2026,"publication_date":"2026-10-23","ids":{"openalex":"https://openalex.org/W7223676583","doi":"https://doi.org/10.48550/arxiv.2610.32320"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.32320","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.32320","pdf_url":"https://arxiv.org/pdf/2610.32320","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/2610.32320","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5075003439","display_name":"Tatsuya Miura","orcid":"https://orcid.org/0000-0002-4454-7660","observed_orcids":["https://orcid.org/0000-0002-4454-7660"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Miura, Tatsuya","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5075003439"],"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.9994999766349792,"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"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.0,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.0,"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/figure-eight-knot","display_name":"figure-eight knot","score":0.9700000286102295},{"id":"https://openalex.org/keywords/planarity","display_name":"planarity","score":0.75},{"id":"https://openalex.org/keywords/twist-knots","display_name":"twist knots","score":0.9139999747276306},{"id":"https://openalex.org/keywords/knot-theory","display_name":"knot theory","score":0.6899999976158142}],"concepts":[{"id":"https://openalex.org/C2779863119","wikidata":"https://www.wikidata.org/wiki/Q6423048","display_name":"Knot (papermaking)","level":2,"score":0.7759000062942505},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6746000051498413},{"id":"https://openalex.org/C2776196297","wikidata":"https://www.wikidata.org/wiki/Q17138781","display_name":"Twist","level":2,"score":0.6539999842643738},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6090999841690063},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.47279998660087585},{"id":"https://openalex.org/C19794929","wikidata":"https://www.wikidata.org/wiki/Q168620","display_name":"Trefoil knot","level":4,"score":0.4578000009059906},{"id":"https://openalex.org/C36794415","wikidata":"https://www.wikidata.org/wiki/Q1862434","display_name":"Knot invariant","level":4,"score":0.44339999556541443},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.42590001225471497},{"id":"https://openalex.org/C143330242","wikidata":"https://www.wikidata.org/wiki/Q849798","display_name":"Knot theory","level":3,"score":0.38960000872612},{"id":"https://openalex.org/C173628569","wikidata":"https://www.wikidata.org/wiki/Q1554293","display_name":"Seifert surface","level":5,"score":0.35920000076293945},{"id":"https://openalex.org/C133776654","wikidata":"https://www.wikidata.org/wiki/Q2381352","display_name":"Fibered knot","level":2,"score":0.34850001335144043},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.337799996137619},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3375999927520752},{"id":"https://openalex.org/C180837527","wikidata":"https://www.wikidata.org/wiki/Q7841270","display_name":"Tricolorability","level":5,"score":0.3310000002384186},{"id":"https://openalex.org/C68260962","wikidata":"https://www.wikidata.org/wiki/Q7200967","display_name":"Planarity testing","level":2,"score":0.32839998602867126},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.32019999623298645},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2711000144481659},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2524999976158142}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.32320","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.32320","pdf_url":"https://arxiv.org/pdf/2610.32320","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.2610.32320","doi":"https://doi.org/10.48550/arxiv.2610.32320","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.32320","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:2610.32320","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.32320","pdf_url":"https://arxiv.org/pdf/2610.32320","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7223676583.pdf","grobid_xml":"https://content.openalex.org/works/W7223676583.grobid-xml"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W1817474533","https://openalex.org/W1906076068","https://openalex.org/W1988986104","https://openalex.org/W2022874746","https://openalex.org/W2023254158","https://openalex.org/W2153292122","https://openalex.org/W2158251088","https://openalex.org/W2796202810","https://openalex.org/W3037941729","https://openalex.org/W3133102727","https://openalex.org/W3163553790","https://openalex.org/W4230108789","https://openalex.org/W4251250309","https://openalex.org/W4385271041","https://openalex.org/W4395029619","https://openalex.org/W4406917268","https://openalex.org/W7147689971","https://openalex.org/W7214009624","https://openalex.org/W7214237926","https://openalex.org/W7223011247"],"related_works":["https://openalex.org/W1614940177","https://openalex.org/W7207716535","https://openalex.org/W2924561345","https://openalex.org/W3108182948","https://openalex.org/W7172487481","https://openalex.org/W2283550438","https://openalex.org/W7099621399","https://openalex.org/W7214009624","https://openalex.org/W2967827925","https://openalex.org/W2915460581"],"abstract_inverted_index":{"We":[0],"prove":[1],"that":[2],"every":[3],"elastic":[4,25],"knot":[5,10],"in":[6],"any":[7],"non-torus":[8],"twist":[9],"class":[11],"is":[12],"not":[13],"planar.":[14],"In":[15],"particular,":[16],"this":[17],"disproves":[18],"a":[19],"numerically":[20],"motivated":[21],"planarity":[22],"conjecture":[23],"for":[24],"figure-eight":[26],"knots.":[27]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-27T00:00:00"}
