{"id":"https://openalex.org/W7225798713","doi":"https://doi.org/10.5281/zenodo.24143810","title":"Toric mosaic numbers of eighteen ten-crossing knots","display_name":"Toric mosaic numbers of eighteen ten-crossing knots","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225798713","doi":"https://doi.org/10.5281/zenodo.24143810"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24143810","doi":"https://doi.org/10.5281/zenodo.24143810","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24143810","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24143810","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5140975118","display_name":"Werner Alois Stanggassinger","orcid":"https://orcid.org/0009-0005-8569-8317","observed_orcids":["https://orcid.org/0009-0005-8569-8317"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Stanggassinger, Werner Alois","raw_affiliation_strings":["Independent (sculptor), Wasserburg am Inn, Germany"],"raw_orcid":"https://orcid.org/0009-0005-8569-8317","affiliations":[{"raw_affiliation_string":"Independent (sculptor), Wasserburg am Inn, Germany","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5140975118"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.9984999895095825,"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":0.9984999895095825,"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":9.999999747378752e-05,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":9.999999747378752e-05,"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":[],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7163000106811523},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5637999773025513},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.5608999729156494},{"id":"https://openalex.org/C186399060","wikidata":"https://www.wikidata.org/wiki/Q959962","display_name":"Diagram","level":2,"score":0.5400000214576721},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.5210000276565552},{"id":"https://openalex.org/C2779863119","wikidata":"https://www.wikidata.org/wiki/Q6423048","display_name":"Knot (papermaking)","level":2,"score":0.4772999882698059},{"id":"https://openalex.org/C45235069","wikidata":"https://www.wikidata.org/wiki/Q278425","display_name":"Table (database)","level":2,"score":0.40049999952316284},{"id":"https://openalex.org/C5052557","wikidata":"https://www.wikidata.org/wiki/Q1528306","display_name":"Toric variety","level":2,"score":0.3815000057220459},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.36410000920295715},{"id":"https://openalex.org/C106131492","wikidata":"https://www.wikidata.org/wiki/Q3072260","display_name":"Filter (signal processing)","level":2,"score":0.30480000376701355},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29760000109672546},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.2646999955177307},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.26019999384880066},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.2533999979496002}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24143810","doi":"https://doi.org/10.5281/zenodo.24143810","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24143810","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.24143811","doi":"https://doi.org/10.5281/zenodo.24143811","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24143811","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24143810","doi":"https://doi.org/10.5281/zenodo.24143810","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24143810","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W2073393217","https://openalex.org/W2588915751","https://openalex.org/W4409760845"],"related_works":[],"abstract_inverted_index":{"Heiney,":[0],"Kipe,":[1],"Pezzimenti,":[2],"Pontes":[3],"and":[4,19,56,68,92,102,145,177,208,275,300,307,332,357,420,432,457,472,504],"Ta":[5],"(arXiv:2504.02265v1;":[6],"Topology":[7],"Appl.":[8],"377":[9],"(2026)":[10],"109657,":[11],"doi:10.1016/j.topol.2025.109657)":[12],"studied":[13],"the":[14,50,73,110,115,127,142,174,180,184,189,195,217,219,227,230,234,241,252,258,261,264,268,277,279,290,296,301,304,318,325,330,334,338,341,344,347,350,355,358,368,373,381,388,391,414,442,448,452,455,458,464,468,470,473,483,488,494,497,502],"toric":[15,54,61,100,134,153,157,186],"mosaic":[16],"number":[17],"mT(K)":[18],"tabulated":[20],"it":[21,249],"for":[22,31,34,72,240],"all":[23],"knots":[24,33,197],"with":[25,149,173,204,257,285,346,441,482],"at":[26,493],"most":[27],"ten":[28],"crossings,":[29],"except":[30,360],"eighteen":[32],"which":[35],"their":[36],"Table":[37],"4":[38,67],"gives":[39],"only":[40,185],"'mT":[41],">=":[42,147],"4,":[43,148],"n/a'.":[44],"From":[45],"a":[46,57,133,152,166,169,286,511],"complete":[47,97],"census":[48],"of":[49,99,109,114,123,168,192,194,206,221,229,233,254,263,270,289,321,337,367,387,450,454,496],"4,934,695,175":[51],"suitably":[52],"connected":[53],"4-mosaics":[55,187],"random":[58],"search":[59],"over":[60],"5-mosaics":[62],"we":[63],"find":[64],"mT(10_129)":[65],"=":[66,70,138],"mT":[69],"5":[71],"other":[74],"seventeen:":[75],"10_17,":[76],"10_52,":[77],"10_76,":[78],"10_81,":[79],"10_86,":[80],"10_88,":[81],"10_90,":[82],"10_96,":[83],"10_97,":[84],"10_98,":[85],"10_105,":[86],"10_110,":[87],"10_114,":[88],"10_115,":[89],"10_119,":[90,207],"10_120":[91],"10_121":[93],"(KnotInfo":[94],"names).":[95],"The":[96,245,396],"censuses":[98],"2-":[101],"3-mosaics":[103],"also":[104,125],"show":[105],"that":[106,163,205,363,393],"three":[107,361],"entries":[108],"arXiv":[111],"version":[112],"v1":[113],"source":[116],"are":[117,202,461,486],"incorrect":[118],"or":[119,198,525],"mutually":[120],"inconsistent,":[121],"two":[122],"them":[124],"in":[126,141,216,293,354,380,439,487],"published":[128,143],"version:":[129],"10_145":[130],"is":[131,171,224,247,402,409],"not":[132,403,500,507],"3-mosaic":[135],"knot,":[136],"mT(5_2)":[137],"3":[139],"(corrected":[140],"version),":[144],"mT(10_132)":[146],"equality":[150],"by":[151,179,327,411,418,477,510],"4-mosaic.":[154],"METHOD:":[155],"every":[156,213,385],"4-mosaic":[158,323],"passing":[159],"an":[160,412],"invariant":[161,351,408],"filter":[162],"cannot":[164],"reject":[165],"diagram":[167,223,232],"target":[170],"traced":[172],"authors'":[175,259,348,369,374,397],"program":[176],"separated":[178],"exact":[181,297],"Jones":[182,190,298,328],"polynomial;":[183,329],"carrying":[188],"polynomial":[191,299],"one":[193],"seventeen":[196],"its":[199,222],"mirror":[200],"image":[201],"256":[203],"they":[209],"represent":[210],"12n_431.":[211],"For":[212],"5-mosaic":[214],"listed":[215],"note":[218,305,389],"complement":[220,228],"isometric":[225],"to":[226,390,463],"KnotInfo":[231],"knot;":[235],"both":[236,294,322],"chiralities":[237],"were":[238,475],"found":[239],"thirteen":[242],"chiral":[243],"knots.":[244],"result":[246],"computer-assisted:":[248],"rests":[250],"on":[251,490],"agreement":[253],"our":[255,365],"trace":[256,370],"code,":[260],"completeness":[262],"enumeration":[265],"(checked":[266],"against":[267],"counts":[269],"Oh,":[271],"Hong,":[272],"Lee,":[273],"Lee":[274],"Yeon),":[276],"filter,":[278],"GPU":[280,339],"kernel":[281],"(its":[282],"hits":[283],"agree":[284],"CPU":[287,335],"recomputation":[288],"whole":[291],"block":[292],"runs),":[295],"identifications.":[302],"CONTENTS:":[303],"(PDF":[306],"LaTeX":[308],"source,":[309],"dated":[310],"19":[311],"October":[312],"2026,":[313],"revised":[314],"11":[315],"November":[316],"2026);":[317],"hit":[319],"lists":[320],"runs;":[324],"separation":[326],"identifications":[331],"witnesses;":[333],"recomputations":[336],"blocks;":[340],"10_132":[342],"census;":[343],"comparisons":[345],"code;":[349],"checks":[352,394],"used":[353],"proof;":[356],"programs,":[359],"files":[362],"contain":[364],"port":[366],"(withheld":[371],"because":[372],"code":[375,398,519],"carries":[376],"no":[377,407],"licence;":[378],"checksums":[379],"README).":[382],"README.md":[383],"maps":[384],"statement":[386],"file":[392],"it.":[395],"(github.com/margekk/toric-mosaics-2025,":[399],"commit":[400],"878037941c3a)":[401],"included.":[404],"HONESTY":[405],"NOTES:":[406],"computed":[410],"AI,":[413],"computations":[415,505],"being":[416],"done":[417],"Python":[419],"C":[421],"programs":[422],"using":[423],"SnapPy":[424],"3.3.2,":[425],"Regina":[426],"7.4,":[427],"SageMath":[428],"10.9,":[429],"database_knotinfo":[430],"2026.10.5":[431],"CuPy":[433],"14.2.0.":[434],"This":[435],"work":[436,456],"was":[437],"developed":[438,476],"collaboration":[440],"AI":[443],"system":[444],"Claude":[445,478],"Code":[446,479],"(Anthropic):":[447],"choice":[449],"problem,":[451],"direction":[453],"computing":[459],"hardware":[460],"due":[462],"author":[465],"(a":[466],"sculptor);":[467],"formalisation,":[469],"algorithms":[471],"proofs":[474],"(Anthropic)":[480],"working":[481],"author.":[484],"Details":[485],"section":[489],"tool":[491],"use":[492],"end":[495],"note.":[498],"STATUS:":[499],"refereed;":[501],"arguments":[503],"have":[506],"been":[508],"checked":[509],"professional":[512],"mathematician.":[513],"Licences:":[514],"text":[515],"CC":[516,528],"BY":[517,529],"4.0;":[518],"GNU":[520],"General":[521],"Public":[522],"License":[523],"v3.0":[524],"later;":[526],"data":[527],"4.0.":[530]},"counts_by_year":[],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-11-12T00:00:00"}
