{"id":"https://openalex.org/W7225309729","doi":"https://doi.org/10.5281/zenodo.23977318","title":"Twist and Writhe of Curves on a Torus: An Exact Formula, Two Families, and the Clifford-Torus Symmetry","display_name":"Twist and Writhe of Curves on a Torus: An Exact Formula, Two Families, and the Clifford-Torus Symmetry","publication_year":2026,"publication_date":"2026-11-05","ids":{"openalex":"https://openalex.org/W7225309729","doi":"https://doi.org/10.5281/zenodo.23977318"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23977318","doi":"https://doi.org/10.5281/zenodo.23977318","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23977318","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.23977318","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5137251060","display_name":"Yunjing Chen","orcid":"https://orcid.org/0009-0008-8147-5089","observed_orcids":["https://orcid.org/0009-0008-8147-5089"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Chen, Yunjing","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0008-8147-5089","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5137251060"],"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":0.9258000254631042,"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":0.9258000254631042,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.010999999940395355,"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/T12504","display_name":"Mathematics and Applications","score":0.006099999882280827,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/torus","display_name":"torus","score":0.5640000104904175},{"id":"https://openalex.org/keywords/clifford-torus","display_name":"Clifford torus","score":0.8600000143051147},{"id":"https://openalex.org/keywords/complete-elliptic-integrals","display_name":"complete elliptic integrals","score":0.6740000247955322},{"id":"https://openalex.org/keywords/loxodromes","display_name":"loxodromes","score":0.8640000224113464},{"id":"https://openalex.org/keywords/stereographic-projection","display_name":"stereographic projection","score":0.7580000162124634},{"id":"https://openalex.org/keywords/flux-surfaces","display_name":"flux surfaces","score":0.4830000102519989},{"id":"https://openalex.org/keywords/toroidal-plasma","display_name":"toroidal plasma","score":0.7720000147819519},{"id":"https://openalex.org/keywords/axisymmetric-magnetic-field","display_name":"axisymmetric magnetic field","score":0.734000027179718},{"id":"https://openalex.org/keywords/linking-number","display_name":"linking number","score":0.5569999814033508},{"id":"https://openalex.org/keywords/permutation-symmetry","display_name":"permutation symmetry","score":0.6050000190734863},{"id":"https://openalex.org/keywords/exact-formulas","display_name":"exact formulas","score":0.414000004529953}],"concepts":[{"id":"https://openalex.org/C2776196297","wikidata":"https://www.wikidata.org/wiki/Q17138781","display_name":"Twist","level":2,"score":0.8248999714851379},{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.7570000290870667},{"id":"https://openalex.org/C78555687","wikidata":"https://www.wikidata.org/wiki/Q778947","display_name":"Writhe","level":3,"score":0.6858999729156494},{"id":"https://openalex.org/C15001198","wikidata":"https://www.wikidata.org/wiki/Q3241394","display_name":"Toroid","level":3,"score":0.6614999771118164},{"id":"https://openalex.org/C98214594","wikidata":"https://www.wikidata.org/wiki/Q850275","display_name":"Conformal map","level":2,"score":0.6194999814033508},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5954999923706055},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.5065000057220459},{"id":"https://openalex.org/C33026886","wikidata":"https://www.wikidata.org/wiki/Q593950","display_name":"Rotational symmetry","level":2,"score":0.46149998903274536},{"id":"https://openalex.org/C203231007","wikidata":"https://www.wikidata.org/wiki/Q192768","display_name":"Paraboloid","level":3,"score":0.42899999022483826},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.4244999885559082},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.42329999804496765},{"id":"https://openalex.org/C54613284","wikidata":"https://www.wikidata.org/wiki/Q1191271","display_name":"Conformal field theory","level":3,"score":0.3946000039577484},{"id":"https://openalex.org/C115260700","wikidata":"https://www.wikidata.org/wiki/Q11408","display_name":"Magnetic field","level":2,"score":0.3781999945640564},{"id":"https://openalex.org/C173848574","wikidata":"https://www.wikidata.org/wiki/Q3507963","display_name":"Conformal symmetry","level":3,"score":0.3752000033855438},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.35409998893737793},{"id":"https://openalex.org/C157479481","wikidata":"https://www.wikidata.org/wiki/Q177831","display_name":"Magnetic flux","level":3,"score":0.33180001378059387},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3100999891757965},{"id":"https://openalex.org/C30707913","wikidata":"https://www.wikidata.org/wiki/Q1545397","display_name":"Minimal surface","level":2,"score":0.2987000048160553},{"id":"https://openalex.org/C119001982","wikidata":"https://www.wikidata.org/wiki/Q866099","display_name":"Stereographic projection","level":2,"score":0.2854999899864197},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.27390000224113464},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.26510000228881836},{"id":"https://openalex.org/C116515362","wikidata":"https://www.wikidata.org/wiki/Q188589","display_name":"Tokamak","level":3,"score":0.26100000739097595},{"id":"https://openalex.org/C68709404","wikidata":"https://www.wikidata.org/wiki/Q1134475","display_name":"Flux (metallurgy)","level":2,"score":0.2547999918460846},{"id":"https://openalex.org/C184670325","wikidata":"https://www.wikidata.org/wiki/Q512604","display_name":"Loop (graph theory)","level":2,"score":0.2540999948978424},{"id":"https://openalex.org/C204795200","wikidata":"https://www.wikidata.org/wiki/Q903282","display_name":"Symmetry breaking","level":2,"score":0.25200000405311584},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.25119999051094055}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23977318","doi":"https://doi.org/10.5281/zenodo.23977318","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23977318","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.23977319","doi":"https://doi.org/10.5281/zenodo.23977319","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23977319","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.23977318","doi":"https://doi.org/10.5281/zenodo.23977318","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23977318","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":2,"referenced_works":["https://openalex.org/W7162666954","https://openalex.org/W7222943648"],"related_works":["https://openalex.org/W1985113682","https://openalex.org/W2167525254","https://openalex.org/W2162954699","https://openalex.org/W2592110604","https://openalex.org/W632995190","https://openalex.org/W2096223954","https://openalex.org/W3123393434","https://openalex.org/W7097410177","https://openalex.org/W2024557039","https://openalex.org/W7075909278"],"abstract_inverted_index":{"Curves":[0],"that":[1,186,208,248],"wind":[2],"on":[3,95,238,268],"a":[4,68,180,199,209,253,288,309,349],"torus":[5,201,271],"occur":[6],"in":[7,11,19,44,64,97,272,339],"DNA":[8,331],"toroidal":[9,16,53],"condensates,":[10,332],"the":[12,23,41,46,52,57,61,73,76,86,89,101,103,107,122,125,131,135,151,154,175,187,204,215,239,249,257,260,264,269,292,322,327,333,341,365,381],"magnetic":[13,193],"surfaces":[14],"of":[15,40,72,92,100,190,256,291,311,330],"plasmas":[17],"and":[18,22,31,169,207,235,262,319,353,361,379],"knotted":[20],"light,":[21],"way":[24],"their":[25],"linking":[26],"number":[27],"divides":[28],"into":[29],"twist":[30,58,104,258,342],"writhe":[32,77,132],"is":[33,138,198,252,287],"usually":[34],"obtained":[35],"numerically.":[36],"For":[37,85],"(p,q)":[38,127],"curves":[39,94],"linear":[42,152],"family,":[43,88,206],"which":[45,340],"poloidal":[47],"angle":[48],"advances":[49],"uniformly":[50],"with":[51,364],"angle,":[54],"we":[55,178,242],"give":[56,179,263],"relative":[59],"to":[60,106,203,295,326],"surface":[62,197],"normal":[63],"closed":[65,273],"form":[66],"as":[67,140,230,236],"complete":[69],"elliptic":[70],"integral":[71],"first":[74],"kind;":[75],"follows":[78],"from":[79],"Wr":[80],"=":[81,111,115,158,173,219,223,276,303,317,370,387],"pq":[82],"\u2212":[83,278,281,313],"Tw.":[84],"conformal":[87,123,205,250],"stereographic":[90],"images":[91],"uniform":[93],"tori":[96,234],"S\u00b3":[98],"(loxodromes":[99],"torus),":[102],"reduces":[105],"elementary":[108],"expression":[109],"Tw":[110],"pq/\u221a(p\u00b2(1\u2212\u03b5\u00b2)+q\u00b2\u03b5\u00b2),":[112],"where":[113],"\u03b5":[114,141],"r/R.":[116],"Two":[117],"consequences":[118],"follow.":[119],"(i)":[120],"In":[121,150],"family":[124,153,251],"exchange":[126,155,227],"\u2194":[128],"(q,p)":[129],"leaves":[130],"unchanged":[133],"provided":[134],"aspect":[136],"ratio":[137],"exchanged":[139],"\u21a6":[142],"\u221a(1\u2212\u03b5\u00b2);":[143],"this":[144],"holds":[145,160],"for":[146,161],"every":[147,162],"\u03b5.":[148],"(ii)":[149],"symmetry":[156],"Wr(p,q)":[157],"Wr(q,p)":[159],"winding":[163,293],"ratio,":[164],"rational":[165],"or":[166],"not,":[167],"if":[168,171,352,355],"only":[170,354],"R/r":[172],"\u221a2,":[174],"Clifford":[176,240,270,366],"torus;":[177,241],"short":[181],"two-sided":[182],"proof.":[183],"We":[184],"show":[185],"field":[188,194],"lines":[189],"any":[191],"axisymmetric":[192],"whose":[195],"flux":[196],"circular-section":[200],"belong":[202],"single":[210],"Shafranov-type":[211],"gradient":[212],"\u03b4":[213,222],"connects":[214],"two":[216],"families":[217],"(\u03b4":[218],"0":[220],"conformal,":[221],"\u2212\u03b5":[224],"linear).":[225],"The":[226,285],"defect":[228,286],"grows":[229],"\u03b4\u00b2":[231],"between":[232],"paired":[233],"\u03b4\u00b3":[237],"prove":[243],"both":[244],"orders":[245],"by":[246,376,380],"showing":[247],"stationary":[254],"point":[255],"along":[259],"bridge,":[261],"leading":[265,296],"cubic":[266],"coefficient":[267],"form,":[274],"c\u2083":[275],"(15/4)(3\u221a2":[277],"4)":[279],"p\u00b3q\u00b3(q\u00b2":[280],"p\u00b2)/(p\u00b2":[282],"+":[283,305,315,385],"q\u00b2)^(7/2).":[284],"smooth":[289],"function":[290],"ratio;":[294],"order":[297],"its":[298],"maximum":[299],"lies":[300],"at":[301,321,368],"\u03bb*":[302],"\u221a((9":[304],"\u221a57)/6)":[306],"\u2248":[307],"1.661,":[308],"root":[310],"3\u03bb\u2074":[312],"9\u03bb\u00b2":[314],"2":[316],"0,":[318],"not":[320],"golden":[323],"ratio.":[324],"Applied":[325],"twisted-toroid":[328],"model":[329],"formula":[334],"predicts":[335],"an":[336],"internal":[337],"layer":[338,350,367],"per":[343],"meridional":[344],"turn":[345],"equals":[346],"one;":[347],"such":[348],"exists":[351],"0.595":[356],"<":[357,359],"k":[358,369],"1,":[360],"it":[362],"coincides":[363],"0.745.":[371],"All":[372],"results":[373],"are":[374],"checked":[375],"independent":[377],"quadratures":[378],"integer":[382],"condition":[383],"|Tw":[384],"Wr|":[386],"pq.":[388]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-11-06T00:00:00"}
