{"id":"https://openalex.org/W7220894359","doi":"https://doi.org/10.48550/arxiv.2610.09386","title":"Non-uniqueness of optimal $1$-embeddings on surfaces up to weak equivalence","display_name":"Non-uniqueness of optimal $1$-embeddings on surfaces up to weak equivalence","publication_year":2026,"publication_date":"2026-10-07","ids":{"openalex":"https://openalex.org/W7220894359","doi":"https://doi.org/10.48550/arxiv.2610.09386"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.09386","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.09386","pdf_url":"https://arxiv.org/pdf/2610.09386","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/2610.09386","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023798510","display_name":"Kengo Enami","orcid":"https://orcid.org/0000-0002-6729-3958","observed_orcids":["https://orcid.org/0000-0002-6729-3958"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Enami, Kengo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5023798510"],"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":0.9409000277519226,"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":0.9409000277519226,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.039400000125169754,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.005499999970197678,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/weak-equivalence","display_name":"weak equivalence","score":0.8130000233650208},{"id":"https://openalex.org/keywords/non-uniqueness","display_name":"non-uniqueness","score":0.8730000257492065},{"id":"https://openalex.org/keywords/closed-surfaces","display_name":"closed surfaces","score":0.9819999933242798},{"id":"https://openalex.org/keywords/torus","display_name":"torus","score":0.9369999766349792},{"id":"https://openalex.org/keywords/connected-sum","display_name":"connected sum","score":0.7609999775886536},{"id":"https://openalex.org/keywords/network-embedding","display_name":"graph embedding","score":0.6850000023841858}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8167999982833862},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.6647999882698059},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5774999856948853},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5206999778747559},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.5116000175476074},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.4959999918937683},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49129998683929443},{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.4878999888896942},{"id":"https://openalex.org/C2776863286","wikidata":"https://www.wikidata.org/wiki/Q17104639","display_name":"Weak equivalence","level":3,"score":0.39640000462532043},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.3783999979496002},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3677999973297119},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.34599998593330383},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.3409999907016754},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.32350000739097595},{"id":"https://openalex.org/C39290043","wikidata":"https://www.wikidata.org/wiki/Q1475760","display_name":"Projective plane","level":3,"score":0.305400013923645},{"id":"https://openalex.org/C157369684","wikidata":"https://www.wikidata.org/wiki/Q34740","display_name":"Genus","level":2,"score":0.27140000462532043},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.26910001039505005},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.267300009727478},{"id":"https://openalex.org/C156103551","wikidata":"https://www.wikidata.org/wiki/Q130998","display_name":"Equivalence relation","level":2,"score":0.2671000063419342}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.09386","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.09386","pdf_url":"https://arxiv.org/pdf/2610.09386","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.2610.09386","doi":"https://doi.org/10.48550/arxiv.2610.09386","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.09386","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:2610.09386","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.09386","pdf_url":"https://arxiv.org/pdf/2610.09386","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":[{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220894359.pdf","grobid_xml":"https://content.openalex.org/works/W7220894359.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4403716963","https://openalex.org/W2810026068","https://openalex.org/W182153543","https://openalex.org/W2305713257","https://openalex.org/W2045737295","https://openalex.org/W2946148018","https://openalex.org/W2982613875","https://openalex.org/W40653327","https://openalex.org/W2263413426","https://openalex.org/W2033336968"],"abstract_inverted_index":{"We":[0],"study":[1],"the":[2,20,31,33,37,58,94],"uniqueness":[3],"of":[4,17],"optimal":[5,43,50,82,91],"$1$-embeddings":[6,92],"on":[7,57],"closed":[8,27],"surfaces":[9,66],"up":[10],"to":[11,64],"weak":[12,74],"equivalence.":[13],"Answering":[14],"a":[15,68,77],"question":[16],"Suzuki":[18],"in":[19],"negative,":[21],"we":[22,79],"show":[23],"that":[24,72],"for":[25],"every":[26],"surface":[28],"other":[29,65],"than":[30],"sphere,":[32],"projective":[34],"plane":[35],"and":[36,60],"Klein":[38],"bottle,":[39],"there":[40],"exists":[41],"an":[42],"$1$-embedded":[44,83],"graph":[45],"admitting":[46],"two":[47],"weakly":[48,89],"inequivalent":[49,90],"$1$-embeddings.":[51],"Our":[52],"construction":[53,71],"is":[54,61],"given":[55],"explicitly":[56],"torus":[59],"then":[62],"extended":[63],"through":[67],"suitable":[69],"connected-sum":[70],"preserves":[73],"inequivalence.":[75],"As":[76],"consequence,":[78],"also":[80],"obtain":[81],"graphs":[84],"with":[85],"exponentially":[86],"many":[87],"pairwise":[88],"as":[93],"genus":[95],"increases.":[96]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-09T00:00:00"}