{"id":"https://openalex.org/W1994909432","doi":"https://doi.org/10.1145/1810959.1810975","title":"Linkless and flat embeddings in 3-space and the unknot problem","display_name":"Linkless and flat embeddings in 3-space and the unknot problem","publication_year":2010,"publication_date":"2010-06-13","ids":{"openalex":"https://openalex.org/W1994909432","doi":"https://doi.org/10.1145/1810959.1810975","mag":"1994909432"},"language":"en","primary_location":{"id":"doi:10.1145/1810959.1810975","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1810959.1810975","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the twenty-sixth annual symposium on Computational geometry","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008961593","display_name":"Ken\u2010ichi Kawarabayashi","orcid":"https://orcid.org/0000-0001-6056-4287"},"institutions":[{"id":"https://openalex.org/I184597095","display_name":"National Institute of Informatics","ror":"https://ror.org/04ksd4g47","country_code":"JP","type":"facility","lineage":["https://openalex.org/I1319490839","https://openalex.org/I184597095","https://openalex.org/I4210158934"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Ken-ichi Kawarabayashi","raw_affiliation_strings":["National Institute of Informatics, Tokyo, Japan","(National Institute of Informatics, Tokyo, Japan)"],"affiliations":[{"raw_affiliation_string":"National Institute of Informatics, Tokyo, Japan","institution_ids":["https://openalex.org/I184597095"]},{"raw_affiliation_string":"(National Institute of Informatics, Tokyo, Japan)","institution_ids":["https://openalex.org/I184597095"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5062011497","display_name":"Stephan Kreutzer","orcid":"https://orcid.org/0009-0002-9409-5356"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Stephan Kreutzer","raw_affiliation_strings":["The University of Oxford, Oxford, United Kingdom","The University of Oxford, Oxford, United Kingdom;"],"affiliations":[{"raw_affiliation_string":"The University of Oxford, Oxford, United Kingdom","institution_ids":["https://openalex.org/I40120149"]},{"raw_affiliation_string":"The University of Oxford, Oxford, United Kingdom;","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5005993212","display_name":"Bojan Mohar","orcid":"https://orcid.org/0000-0002-7408-6148"},"institutions":[{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Bojan Mohar","raw_affiliation_strings":["Simon Fraser University, Burnaby, Canada","( Simon Fraser University Burnaby Canada )"],"affiliations":[{"raw_affiliation_string":"Simon Fraser University, Burnaby, Canada","institution_ids":["https://openalex.org/I18014758"]},{"raw_affiliation_string":"( Simon Fraser University Burnaby Canada )","institution_ids":["https://openalex.org/I18014758"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5008961593"],"corresponding_institution_ids":["https://openalex.org/I184597095"],"apc_list":null,"apc_paid":null,"fwci":0.669,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.69047343,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":null,"issue":null,"first_page":"97","last_page":"106"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9988999962806702,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9988999962806702,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.9987999796867371,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9980999827384949,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/unknot","display_name":"Unknot","score":0.9423716068267822},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.735183596611023},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.6814104914665222},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6692593097686768},{"id":"https://openalex.org/keywords/converse","display_name":"Converse","score":0.5881364345550537},{"id":"https://openalex.org/keywords/book-embedding","display_name":"Book embedding","score":0.5369863510131836},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.5020153522491455},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.48831459879875183},{"id":"https://openalex.org/keywords/planar-graph","display_name":"Planar graph","score":0.4681214392185211},{"id":"https://openalex.org/keywords/knot","display_name":"Knot (papermaking)","score":0.4441972076892853},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.41396060585975647},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.4137447774410248},{"id":"https://openalex.org/keywords/chordal-graph","display_name":"Chordal graph","score":0.2760944962501526},{"id":"https://openalex.org/keywords/1-planar-graph","display_name":"1-planar graph","score":0.20126521587371826},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.19125598669052124},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07249900698661804}],"concepts":[{"id":"https://openalex.org/C2780286412","wikidata":"https://www.wikidata.org/wiki/Q1188344","display_name":"Unknot","level":3,"score":0.9423716068267822},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.735183596611023},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.6814104914665222},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6692593097686768},{"id":"https://openalex.org/C2776809875","wikidata":"https://www.wikidata.org/wiki/Q1375963","display_name":"Converse","level":2,"score":0.5881364345550537},{"id":"https://openalex.org/C180222743","wikidata":"https://www.wikidata.org/wiki/Q4942980","display_name":"Book embedding","level":5,"score":0.5369863510131836},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.5020153522491455},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48831459879875183},{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.4681214392185211},{"id":"https://openalex.org/C2779863119","wikidata":"https://www.wikidata.org/wiki/Q6423048","display_name":"Knot (papermaking)","level":2,"score":0.4441972076892853},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.41396060585975647},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.4137447774410248},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.2760944962501526},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.20126521587371826},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.19125598669052124},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07249900698661804},{"id":"https://openalex.org/C42360764","wikidata":"https://www.wikidata.org/wiki/Q83588","display_name":"Chemical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1810959.1810975","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1810959.1810975","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the twenty-sixth annual symposium on Computational geometry","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":57,"referenced_works":["https://openalex.org/W132145278","https://openalex.org/W134472367","https://openalex.org/W142273131","https://openalex.org/W174566906","https://openalex.org/W594463866","https://openalex.org/W1489922475","https://openalex.org/W1520619927","https://openalex.org/W1521990134","https://openalex.org/W1529230708","https://openalex.org/W1589241842","https://openalex.org/W1663305996","https://openalex.org/W1964700557","https://openalex.org/W1975433230","https://openalex.org/W1982180670","https://openalex.org/W1983694630","https://openalex.org/W1987374783","https://openalex.org/W2005079828","https://openalex.org/W2024931493","https://openalex.org/W2026275056","https://openalex.org/W2038713190","https://openalex.org/W2042932000","https://openalex.org/W2047577363","https://openalex.org/W2052259760","https://openalex.org/W2054639744","https://openalex.org/W2057826895","https://openalex.org/W2059372025","https://openalex.org/W2059563583","https://openalex.org/W2060765530","https://openalex.org/W2061980451","https://openalex.org/W2064796716","https://openalex.org/W2065805017","https://openalex.org/W2067016370","https://openalex.org/W2068497209","https://openalex.org/W2082687277","https://openalex.org/W2085348754","https://openalex.org/W2091659372","https://openalex.org/W2092904196","https://openalex.org/W2094004458","https://openalex.org/W2097289283","https://openalex.org/W2099817679","https://openalex.org/W2115607023","https://openalex.org/W2122764799","https://openalex.org/W2132750992","https://openalex.org/W2133385765","https://openalex.org/W2142607374","https://openalex.org/W2145145566","https://openalex.org/W2156760067","https://openalex.org/W2161714963","https://openalex.org/W2165303918","https://openalex.org/W2169534903","https://openalex.org/W2318517541","https://openalex.org/W2401216365","https://openalex.org/W2798588639","https://openalex.org/W2911779984","https://openalex.org/W2963875567","https://openalex.org/W4210246355","https://openalex.org/W4319589071"],"related_works":["https://openalex.org/W2185062889","https://openalex.org/W2990852788","https://openalex.org/W1986542256","https://openalex.org/W2063120809","https://openalex.org/W2954828522","https://openalex.org/W2036908020","https://openalex.org/W2052310372","https://openalex.org/W2071261821","https://openalex.org/W2029772672","https://openalex.org/W1852912860"],"abstract_inverted_index":{"We":[0,137],"consider":[1,139],"piecewise":[2],"linear":[3],"embeddings":[4,128,146],"of":[5,18,28,58,71,107],"graphs":[6,60],"in":[7,44,61,85,147],"3-space":[8],"\u211c3.":[9],"Such":[10],"an":[11,99],"embbeding":[12],"is":[13,82,129,134],"linkless":[14,42],"if":[15],"every":[16,104,126],"pair":[17],"disjoint":[19],"cycles":[20],"forms":[21],"a":[22,38,41,55,69,80,93,111],"trivial":[23],"link":[24],"(in":[25],"the":[26,132,140],"sense":[27],"knot":[29],"theory).":[30],"Robertson,":[31],"Seymour":[32],"and":[33,47,73,88],"Thomas":[34],"[47]":[35],"showed":[36,78],"that":[37,79,102],"graph":[39,81],"has":[40],"embedding":[43,95,100],"\u211c3":[45,86,116],"if,":[46,49,87,90],"only":[48,89],"it":[50,91],"does":[51],"not":[52,135],"contain":[53],"as":[54],"minor":[56],"any":[57],"seven":[59],"Petersen's":[62],"family":[63],"(graphs":[64],"obtained":[65],"from":[66],"K6":[67],"by":[68],"series":[70],"Y\u0394":[72],"\u0394Y":[74],"operations).":[75],"They":[76],"also":[77],"linklessly":[83],"embeddable":[84],"admits":[92],"flat":[94,127],"into":[96],"\u211c3,":[97],"i.e.":[98],"such":[101],"for":[103],"cycle":[105],"C":[106],"G":[108,120],"there":[109],"exists":[110],"closed":[112],"2-disk":[113],"D":[114,118],"\u2286":[115],"with":[117,145],"\u2229":[119],"=":[121,123],"\u2202D":[122],"C.":[124],"Clearly,":[125],"linkless,":[130],"but":[131],"converse":[133],"true.":[136],"first":[138],"following":[141],"algorithmic":[142],"problem":[143],"associated":[144],"\u211c3:":[148]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}