{"id":"https://openalex.org/W2950461635","doi":"https://doi.org/10.20382/jocg.v10i1a15","title":"Lombardi Drawings of Knots and Links","display_name":"Lombardi Drawings of Knots and Links","publication_year":2017,"publication_date":"2017-08-31","ids":{"openalex":"https://openalex.org/W2950461635","doi":"https://doi.org/10.20382/jocg.v10i1a15","mag":"2950461635"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:1708.09819","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1708.09819","pdf_url":"https://arxiv.org/pdf/1708.09819","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"","raw_type":"text"},"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1708.09819","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029516206","display_name":"Philipp Kindermann","orcid":"https://orcid.org/0000-0001-5764-7719"},"institutions":[{"id":"https://openalex.org/I25974101","display_name":"University of W\u00fcrzburg","ror":"https://ror.org/00fbnyb24","country_code":"DE","type":"education","lineage":["https://openalex.org/I25974101"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Kindermann, Philipp","raw_affiliation_strings":["University of Wurzburg#TAB#"],"affiliations":[{"raw_affiliation_string":"University of Wurzburg#TAB#","institution_ids":["https://openalex.org/I25974101"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5003147292","display_name":"Stephen Kobourov","orcid":"https://orcid.org/0000-0002-0477-2724"},"institutions":[{"id":"https://openalex.org/I138006243","display_name":"University of Arizona","ror":"https://ror.org/03m2x1q45","country_code":"US","type":"education","lineage":["https://openalex.org/I138006243"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kobourov, Stephen","raw_affiliation_strings":["University of Arizona"],"affiliations":[{"raw_affiliation_string":"University of Arizona","institution_ids":["https://openalex.org/I138006243"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5112453554","display_name":"Maarten L\u00f6ffler","orcid":null},"institutions":[{"id":"https://openalex.org/I193662353","display_name":"Utrecht University","ror":"https://ror.org/04pp8hn57","country_code":"NL","type":"education","lineage":["https://openalex.org/I193662353"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"L\u00f6ffler, Maarten","raw_affiliation_strings":["Utrecht University"],"affiliations":[{"raw_affiliation_string":"Utrecht University","institution_ids":["https://openalex.org/I193662353"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071390048","display_name":"Martin N\u00f6llenburg","orcid":"https://orcid.org/0000-0003-0454-3937"},"institutions":[{"id":"https://openalex.org/I145847075","display_name":"TU Wien","ror":"https://ror.org/04d836q62","country_code":"AT","type":"education","lineage":["https://openalex.org/I145847075"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"N\u00f6llenburg, Martin","raw_affiliation_strings":["Vienna University of Technology"],"affiliations":[{"raw_affiliation_string":"Vienna University of Technology","institution_ids":["https://openalex.org/I145847075"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055531990","display_name":"Andr\u00e9 Schulz","orcid":"https://orcid.org/0000-0002-2134-4852"},"institutions":[{"id":"https://openalex.org/I120691247","display_name":"University of Hagen","ror":"https://ror.org/04tkkr536","country_code":"DE","type":"education","lineage":["https://openalex.org/I120691247"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Schulz, Andr\u00e9","raw_affiliation_strings":["Univ. of Hagen"],"affiliations":[{"raw_affiliation_string":"Univ. of Hagen","institution_ids":["https://openalex.org/I120691247"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073288056","display_name":"Birgit Vogtenhuber","orcid":"https://orcid.org/0000-0002-7166-4467"},"institutions":[{"id":"https://openalex.org/I4092182","display_name":"Graz University of Technology","ror":"https://ror.org/00d7xrm67","country_code":"AT","type":"education","lineage":["https://openalex.org/I4092182"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Vogtenhuber, Birgit","raw_affiliation_strings":["Graz University of Technology,#TAB#"],"affiliations":[{"raw_affiliation_string":"Graz University of Technology,#TAB#","institution_ids":["https://openalex.org/I4092182"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5029516206"],"corresponding_institution_ids":["https://openalex.org/I25974101"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.3650822,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9980000257492065,"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"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9980000257492065,"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"}},{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.9975000023841858,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.989799976348877,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/knot","display_name":"Knot (papermaking)","score":0.6985520124435425},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5607513785362244},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5544021129608154},{"id":"https://openalex.org/keywords/offset","display_name":"Offset (computer science)","score":0.5012807846069336},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.49282410740852356},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.44693124294281006},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4233365058898926},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1419905126094818}],"concepts":[{"id":"https://openalex.org/C2779863119","wikidata":"https://www.wikidata.org/wiki/Q6423048","display_name":"Knot (papermaking)","level":2,"score":0.6985520124435425},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5607513785362244},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5544021129608154},{"id":"https://openalex.org/C175291020","wikidata":"https://www.wikidata.org/wiki/Q1156822","display_name":"Offset (computer science)","level":2,"score":0.5012807846069336},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.49282410740852356},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.44693124294281006},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4233365058898926},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1419905126094818},{"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"pmh:oai:arXiv.org:1708.09819","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1708.09819","pdf_url":"https://arxiv.org/pdf/1708.09819","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"","raw_type":"text"},{"id":"pmh:oai:ojs.library.carleton.ca:article/3081","is_oa":true,"landing_page_url":"https://jocg.org/index.php/jocg/article/view/3081","pdf_url":"https://jocg.org/index.php/jocg/article/view/3081","source":{"id":"https://openalex.org/S4306400962","display_name":"Journal of Computational Geometry (Carleton University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I67031392","host_organization_name":"Carleton University","host_organization_lineage":["https://openalex.org/I67031392"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Computational Geometry; Vol. 10 No. 1 (2019); 444\u2013476","raw_type":"info:eu-repo/semantics/article"},{"id":"mag:2950461635","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1708.09819.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:uu:oai:dspace.library.uu.nl:1874/396408","is_oa":true,"landing_page_url":"https://dspace.library.uu.nl/handle/1874/396408","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Proceedings of the 25th Symposium on Graph Drawing","raw_type":"info:eu-repo/semantics/bookpart"},{"id":"doi:10.48550/arxiv.1708.09819","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1708.09819","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"raw_source_name":null,"raw_type":"article"},{"id":"doi:10.20382/jocg.v10i1a15","is_oa":true,"landing_page_url":"https://doi.org/10.20382/jocg.v10i1a15","pdf_url":null,"source":{"id":"https://openalex.org/S4306400962","display_name":"Journal of Computational Geometry (Carleton University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I67031392","host_organization_name":"Carleton University","host_organization_lineage":["https://openalex.org/I67031392"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1708.09819","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1708.09819","pdf_url":"https://arxiv.org/pdf/1708.09819","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W421616731","https://openalex.org/W634862121","https://openalex.org/W1503792772","https://openalex.org/W1533187596","https://openalex.org/W1605569140","https://openalex.org/W1607742244","https://openalex.org/W1847044884","https://openalex.org/W1990695568","https://openalex.org/W1997530681","https://openalex.org/W2012855423","https://openalex.org/W2027638286","https://openalex.org/W2034666435","https://openalex.org/W2080794274","https://openalex.org/W2082388593","https://openalex.org/W2109095183","https://openalex.org/W2316343226","https://openalex.org/W3106094302"],"related_works":["https://openalex.org/W2751467574","https://openalex.org/W2010889111","https://openalex.org/W840724662","https://openalex.org/W2964307244","https://openalex.org/W1548041420","https://openalex.org/W2795920866","https://openalex.org/W2029497772","https://openalex.org/W1617596005","https://openalex.org/W3029906806","https://openalex.org/W3099343474","https://openalex.org/W2952217229","https://openalex.org/W2053285980","https://openalex.org/W1603862932","https://openalex.org/W2027638286","https://openalex.org/W2023254158","https://openalex.org/W84799921","https://openalex.org/W2088666641","https://openalex.org/W3098486263","https://openalex.org/W3195986691","https://openalex.org/W3022859524"],"abstract_inverted_index":{"Knot":[0],"and":[1,57,72,113],"link":[2],"diagrams":[3,31],"are":[4,32,39,125],"projections":[5,47],"of":[6,34,64,96,110,127],"one":[7],"or":[8],"more":[9,19],"3-dimensional":[10],"simple":[11],"closed":[12],"curves":[13,42,51],"into":[14],"$R^2$,":[15],"such":[16],"that":[17,78,100,115,136],"no":[18],"than":[20],"two":[21,108,128],"points":[22],"project":[23],"to":[24],"the":[25,62,88,147],"same":[26],"point":[27],"in":[28,43,52],"$R^2$.":[29],"These":[30],"drawings":[33,67,112],"4-regular":[35,97],"plane":[36,84,98,120],"multigraphs.":[37],"Knots":[38],"typically":[40],"smooth":[41,50],"$R^3$,":[44],"so":[45],"their":[46],"should":[48],"be":[49,140],"$R^2$":[53],"with":[54],"good":[55],"continuity":[56],"large":[58,94],"crossing":[59],"angles:":[60],"exactly":[61],"properties":[63],"Lombardi":[65,85,103,111,143],"graph":[66],"(defined":[68],"by":[69,151],"circular-arc":[70],"edges":[71,124],"perfect":[73],"angular":[74,148,155],"resolution).":[75],"We":[76,105],"show":[77,114],"several":[79],"knots":[80],"do":[81,101],"not":[82],"allow":[83],"drawings.":[86,104],"On":[87],"other":[89],"hand,":[90],"we":[91],"identify":[92],"a":[93,119,160],"class":[95],"multigraphs":[99],"have":[102],"then":[106],"study":[107],"relaxations":[109],"every":[116,132],"knot":[117,133],"admits":[118],"2-Lombardi":[121],"drawing":[122,144],"(where":[123],"composed":[126],"circular":[129],"arcs).":[130],"Further,":[131],"is":[134],"near-Lombardi,":[135],"is,":[137],"it":[138],"can":[139],"drawn":[141],"as":[142],"when":[145],"relaxing":[146],"resolution":[149],"requirement":[150],"an":[152],"arbitrary":[153],"small":[154],"offset":[156],"$\\varepsilon$,":[157],"while":[158],"maintaining":[159],"$180^\\circ$":[161],"angle":[162],"between":[163],"opposite":[164],"edges.":[165]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-04-04T08:04:53.788161","created_date":"2025-10-10T00:00:00"}
