{"id":"https://openalex.org/W1626240448","doi":"https://doi.org/10.37236/2366","title":"Reflexive Polytopes of Higher Index and the Number 12","display_name":"Reflexive Polytopes of Higher Index and the Number 12","publication_year":2012,"publication_date":"2012-07-19","ids":{"openalex":"https://openalex.org/W1626240448","doi":"https://doi.org/10.37236/2366","mag":"1626240448"},"language":"en","primary_location":{"id":"doi:10.37236/2366","is_oa":true,"landing_page_url":"https://doi.org/10.37236/2366","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v19i3p9/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v19i3p9/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5059138304","display_name":"Alexander Kasprzyk","orcid":"https://orcid.org/0000-0003-2340-5257"},"institutions":[{"id":"https://openalex.org/I47508984","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10","country_code":"GB","type":"education","lineage":["https://openalex.org/I47508984"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Alexander M. Kasprzyk","raw_affiliation_strings":["Imperial College London","Imperial College London,#TAB#"],"affiliations":[{"raw_affiliation_string":"Imperial College London","institution_ids":["https://openalex.org/I47508984"]},{"raw_affiliation_string":"Imperial College London,#TAB#","institution_ids":["https://openalex.org/I47508984"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5084262984","display_name":"Benjamin Nill","orcid":"https://orcid.org/0000-0001-5671-8271"},"institutions":[{"id":"https://openalex.org/I58956616","display_name":"Case Western Reserve University","ror":"https://ror.org/051fd9666","country_code":"US","type":"education","lineage":["https://openalex.org/I58956616"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Benjamin Nill","raw_affiliation_strings":["Case Western Reserve University","Case Western Reserve University ,"],"affiliations":[{"raw_affiliation_string":"Case Western Reserve University","institution_ids":["https://openalex.org/I58956616"]},{"raw_affiliation_string":"Case Western Reserve University ,","institution_ids":["https://openalex.org/I58956616"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5059138304"],"corresponding_institution_ids":["https://openalex.org/I47508984"],"apc_list":null,"apc_paid":null,"fwci":1.2574,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.75838878,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"19","issue":"3","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9783999919891357,"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/T12504","display_name":"Mathematics and Applications","score":0.9783999919891357,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9750999808311462,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9139999747276306,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/polytope","display_name":"Polytope","score":0.8441891074180603},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7477800250053406},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7072980999946594},{"id":"https://openalex.org/keywords/reflexivity","display_name":"Reflexivity","score":0.6485093832015991},{"id":"https://openalex.org/keywords/polygon","display_name":"Polygon (computer graphics)","score":0.5903106331825256},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.46487975120544434},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.45957136154174805},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.43533435463905334},{"id":"https://openalex.org/keywords/isomorphism","display_name":"Isomorphism (crystallography)","score":0.4133286774158478},{"id":"https://openalex.org/keywords/index","display_name":"Index (typography)","score":0.41002780199050903},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36054229736328125},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1387476623058319},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09970664978027344}],"concepts":[{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.8441891074180603},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7477800250053406},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7072980999946594},{"id":"https://openalex.org/C13200473","wikidata":"https://www.wikidata.org/wiki/Q2747831","display_name":"Reflexivity","level":2,"score":0.6485093832015991},{"id":"https://openalex.org/C190694206","wikidata":"https://www.wikidata.org/wiki/Q3276654","display_name":"Polygon (computer graphics)","level":3,"score":0.5903106331825256},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.46487975120544434},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.45957136154174805},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.43533435463905334},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.4133286774158478},{"id":"https://openalex.org/C2777382242","wikidata":"https://www.wikidata.org/wiki/Q6017816","display_name":"Index (typography)","level":2,"score":0.41002780199050903},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36054229736328125},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1387476623058319},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09970664978027344},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C136764020","wikidata":"https://www.wikidata.org/wiki/Q466","display_name":"World Wide Web","level":1,"score":0.0},{"id":"https://openalex.org/C144024400","wikidata":"https://www.wikidata.org/wiki/Q21201","display_name":"Sociology","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C126042441","wikidata":"https://www.wikidata.org/wiki/Q1324888","display_name":"Frame (networking)","level":2,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C115624301","wikidata":"https://www.wikidata.org/wiki/Q895901","display_name":"Crystal structure","level":2,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C36289849","wikidata":"https://www.wikidata.org/wiki/Q34749","display_name":"Social science","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/2366","is_oa":true,"landing_page_url":"https://doi.org/10.37236/2366","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v19i3p9/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1107.4945","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1107.4945","pdf_url":"https://arxiv.org/pdf/1107.4945","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":null,"raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/2366","is_oa":true,"landing_page_url":"https://doi.org/10.37236/2366","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v19i3p9/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1445980183","display_name":null,"funder_award_id":"Platform grant EP/I019111/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G2628764907","display_name":"Imperial College London Mathematics Platform Grant","funder_award_id":"EP/I019111/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G3330119427","display_name":null,"funder_award_id":"EPSRC","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6000271606","display_name":"Extremal Laurent Polynomials","funder_award_id":"EP/I008128/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G6769360737","display_name":null,"funder_award_id":"Mathematics Platform grant EP/I019111/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G801152762","display_name":null,"funder_award_id":"1203162","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G837760560","display_name":null,"funder_award_id":"DMS 1203162","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8654818507","display_name":null,"funder_award_id":"Mathematics","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G8700811484","display_name":null,"funder_award_id":"EP/I008128/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G8895942654","display_name":null,"funder_award_id":"Imperial College","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320320283","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10"},{"id":"https://openalex.org/F4320320966","display_name":"University of Sydney","ror":"https://ror.org/0384j8v12"},{"id":"https://openalex.org/F4320324521","display_name":"Freie Universit\u00e4t Berlin","ror":null},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1626240448.pdf","grobid_xml":"https://content.openalex.org/works/W1626240448.grobid-xml"},"referenced_works_count":36,"referenced_works":["https://openalex.org/W1504534822","https://openalex.org/W1512454594","https://openalex.org/W1521877949","https://openalex.org/W1584105846","https://openalex.org/W1616836920","https://openalex.org/W1625610113","https://openalex.org/W1789739501","https://openalex.org/W1796115931","https://openalex.org/W1818846042","https://openalex.org/W1844594000","https://openalex.org/W1986040477","https://openalex.org/W2001252680","https://openalex.org/W2012395617","https://openalex.org/W2014380411","https://openalex.org/W2015438979","https://openalex.org/W2021760526","https://openalex.org/W2050266449","https://openalex.org/W2055622362","https://openalex.org/W2075015671","https://openalex.org/W2077055195","https://openalex.org/W2089002882","https://openalex.org/W2116083462","https://openalex.org/W2123608978","https://openalex.org/W2155718882","https://openalex.org/W2165868045","https://openalex.org/W2608329354","https://openalex.org/W2951484734","https://openalex.org/W2963800692","https://openalex.org/W2993407187","https://openalex.org/W3098822487","https://openalex.org/W3106473208","https://openalex.org/W4240283984","https://openalex.org/W4243550821","https://openalex.org/W4301042554","https://openalex.org/W4394664565","https://openalex.org/W6684566061"],"related_works":["https://openalex.org/W4293241344","https://openalex.org/W2361927408","https://openalex.org/W4287022798","https://openalex.org/W2154141401","https://openalex.org/W2884061808","https://openalex.org/W1585992543","https://openalex.org/W2901329027","https://openalex.org/W2053943947","https://openalex.org/W2215744270","https://openalex.org/W2314484939"],"abstract_inverted_index":{"We":[0,115],"introduce":[1],"reflexive":[2,16,38,70],"polytopes":[3,23],"of":[4,11,14,18,43,56,72,87],"index":[5,19,61,74],"$l$":[6],"as":[7,26],"a":[8,15,41,83],"natural":[9],"generalisation":[10],"the":[12,44,76,95],"notion":[13],"polytope":[17],"$1$.":[20],"These":[21],"$l$-reflexive":[22,57,109],"also":[24,104],"appear":[25],"dual":[27],"pairs.":[28],"In":[29],"dimension":[30],"two":[31],"we":[32,66],"show":[33,67],"that":[34,68],"they":[35],"arise":[36],"from":[37],"polygons":[39,58,89],"via":[40],"change":[42],"underlying":[45],"lattice.":[46],"This":[47,81],"allows":[48],"us":[49],"to":[50,60],"efficiently":[51],"classify":[52],"all":[53],"isomorphism":[54],"classes":[55],"up":[59],"$200$.":[62],"As":[63],"another":[64],"application,":[65],"any":[69],"polygon":[71],"arbitrary":[73],"satisfies":[75],"famous":[77],"\"number":[78],"$12$\"":[79],"property.":[80],"is":[82],"new,":[84],"infinite":[85],"class":[86],"lattice":[88],"possessing":[90],"this":[91],"property,":[92],"and":[93,121],"extends":[94],"previously":[96],"known":[97],"sixteen":[98],"instances.":[99],"The":[100],"number":[101],"$12$":[102],"property":[103],"holds":[105],"more":[106],"generally":[107],"for":[108],"non-convex":[110],"or":[111],"self-intersecting":[112],"polygonal":[113],"loops.":[114],"conclude":[116],"by":[117],"discussing":[118],"higher-dimensional":[119],"examples":[120],"open":[122],"questions.":[123]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
