{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-0VYU4G4I/d56b440d-97cf-4a9f-bda3-98fc7c5f9f62/PDF","dcterms:extent":"372 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-0VYU4G4I/2674ba7c-e25c-48aa-b060-631f59be1754/TEXT","dcterms:extent":"33 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-0VYU4G4I","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2015","dc:creator":"Hidalgo, Rubén A.","dc:format":[{"@xml:lang":"sl","#text":"številka:2"},{"@xml:lang":"sl","#text":"letnik:8"},{"@xml:lang":"sl","#text":"str. 275-289"}],"dc:identifier":["COBISSID:17376601","ISSN:1855-3966","URN:URN:NBN:SI:doc-0VYU4G4I"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"en","#text":"algebraic curve"},{"@xml:lang":"sl","#text":"algebraična krivulja"},{"@xml:lang":"en","#text":"dessin d'enfant"},{"@xml:lang":"sl","#text":"otroška risba"},{"@xml:lang":"en","#text":"Riemann surface"},{"@xml:lang":"sl","#text":"Riemannova ploskev"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Edmonds maps on the Fricke-Macbeath curve|"},"dc:description":[{"@xml:lang":"sl","#text":"In 1985, L. D. James and G. A. Jones proved that the complete graph ?$K_n$? defines a clean dessin d'enfant (the bipartite graph is given by taking as the black vertices the vertices of ?$K_n$? and the white vertices as middle points of edges) if and only if ?$n = p^e$?, where ?$p$? is a prime. Later, in 2010, G. A. Jones, M. Streit and J. Wolfart computed the minimal field of definition of them. The minimal genus ?$g > 1$? of these types of clean dessins d'enfant is ?$g = 7$?, obtained for ?$p = 2$? and ?$e = 3$?. In that case, there are exactly two such clean dessins d'enfant (previously known as Edmonds maps), both of them defining the Fricke-Macbeath curve (the only Hurwitz curve of genus 7) and both forming a chiral pair. The uniqueness of the Fricke-Macbeath curve asserts that it is definable over ?$\\mathbb{Q}$?, but both Edmonds maps cannot be defined over ?$\\mathbb{Q}$?; in fact they have as minimal field of definition the quadratic field ?$\\mathbb{Q}(\\sqrt{- 7})$?. It seems that no explicit models for the Edmonds maps over ?$\\mathbb{Q}(\\sqrt{- 7})$? are written in the literature. In this paper we start with an explicit model ?$X$? for the Fricke-Macbeath curve provided by Macbeath, which is defined over ?$\\mathbb{Q}(e^{2\\pi i/7})$?, and we construct an explicit birational isomorphism ?$L \\colon X \\to Z$?, where ?$Z$? is defined over ?$\\mathbb{Q}(\\sqrt{- 7})$?, so that both Edmonds maps are also defined over that field"},{"@xml:lang":"sl","#text":"Leta 1985 so L. D. James in G. A. Jones dokazali, da polni graf ?$K_n$? definira čisto otroško risbo (za črna vozlišča dvodelnega grafa vzamemo vozlišča grafa ?$K_n$?, za bela vozlišča pa središčne točke njegovih povezav) če in samo če je ?$n = p^e$?, kjer je ?$p$? praštevilo. Kasneje, leta 2010, so G. A. Jones, M. Streit in J. Wolfart izračunali minimalno definicijsko območje takšnih čistih otroških risb. Minimalni rod ?$g > 1$? teh tipov čistih otroških risb je ?$g = 7$?, dobljen pa je pri ?$p = 2$? in ?$e = 3$?. V tem primeru obstajata natanko dve takšni čisti otroški risbi, (doslej znani pod imenom Edmondsova zemljevida), ki določata t.i. Fricke-Macbeathovo krivuljo (edino Hurwitzovo krivuljo roda 7) in tvorita kiralni par. Enoličnost Fricke-Macbeathove krivulje zagotavlja, da jo je mogoče definirati nad ?$\\mathbb{Q}$?, toda oba Edmondsova zemljevida ne moreta biti definirana nad ?$\\mathbb{Q}$?; v resnici je njuno minimalno definicijsko območje kvadratni obseg ?$\\mathbb{Q}(\\sqrt{- 7})$?. Zdi se, da v literaturi niso zabeleženi nobeni eksplicitni modeli Edmondsovih zemljevidov nad ?$\\mathbb{Q}(\\sqrt{- 7})$?. V tem članku vzamemo za izhodišče ekspliciten Macbeathov model ?$X$? za Fricke-Macbeathovo krivuljo, ki je definirana nad ?$\\mathbb{Q}(e^{2\\pi i/7})$?, potem pa konstruiramo ekspliciten biracionalen izomorfizem ?$L \\colon X \\to Z$?, kjer je ?$Z$? definiran nad ?$\\mathbb{Q}(\\sqrt{- 7})$?, nad istim obsegom pa sta definirana tudi oba Edmondsova zemljevida"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-0VYU4G4I","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-0VYU4G4I"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-0VYU4G4I/d56b440d-97cf-4a9f-bda3-98fc7c5f9f62/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-0VYU4G4I/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-0VYU4G4I"}}}}