The slack model in the study of polytopes

In this thesis we present some of the applications of studying polytopes via their slack matrices and slack ideals. We prove that McMullen's operations on polytopes, which includes the join, the vertex sum, the vertex splitting and their dual operations, preserve graphicality in the same way it...

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Autores:
Torres Chaves, Juan Camilo
Tipo de recurso:
Doctoral thesis
Fecha de publicación:
2020
Institución:
Universidad de los Andes
Repositorio:
Séneca: repositorio Uniandes
Idioma:
eng
OAI Identifier:
oai:repositorio.uniandes.edu.co:1992/51154
Acceso en línea:
http://hdl.handle.net/1992/51154
Palabra clave:
Politopos
Geometría algebraica
Matemáticas
Matemáticas
Rights
openAccess
License
http://creativecommons.org/licenses/by-nc-sa/4.0/
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spelling Al consultar y hacer uso de este recurso, está aceptando las condiciones de uso establecidas por los autores.http://creativecommons.org/licenses/by-nc-sa/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Bogart, Tristram47bee2c6-53e2-4211-831b-906b5a30b6ac600Gouveia, Joáof51b04da-6be4-4317-bd95-836d49b99c6c500Torres Chaves, Juan Camilo29797500Padrol, ArnauSchurmann, AchillVelasco Gregory, Mauricio Fernando2021-08-10T18:13:02Z2021-08-10T18:13:02Z2020http://hdl.handle.net/1992/5115410.57784/1992/5115423522.pdfinstname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/In this thesis we present some of the applications of studying polytopes via their slack matrices and slack ideals. We prove that McMullen's operations on polytopes, which includes the join, the vertex sum, the vertex splitting and their dual operations, preserve graphicality in the same way it was known it preserves projective uniqueness. We use this result to identify a large class of projectively unique order polytopes; namely, we prove that every ranked finite poset with no 3-antichain has a graphic and thus projectively unique order polytope. We give a complete characterization of complex psd-minimal polygons; we prove that the complex psd-minimal polygons are precisely the triangles, the quadrialterals and a special class of hexagons which we call Pappus' hexagons. Using this, it can be seen that complex psd-minimal 3-polytopes must have vertices of degree 3, 4 or 6 and facets with 3, 4 or 6 sides. We identify all the combinatorial classes of 3-polytopes...En esta tesis presentamos algunas aplicaciones de estudiar politopos a través de sus matrices de holgura e ideales de holgura. Demostramos que las operaciones de McMullen sobre politopos, las cuales incluyen la unión, la suma sobre un vértice, la ruptura de un vértice y sus operaciones duales, preservan la graficalidad de la misma manera que se sabía preservaban la unicidad projectiva. Usamos este resultado para identificar una clase grande de politopos de orden proyectivamente únicos; de manera más precisa, demostramos que todo poset finito con rango y sin 3-anticadenas tiene un politopo de orden gráfico y por ende proyectivamente único. Damos una caracterización completa de los polígonos psd-minimales complejos; demostramos que los polígonos psd-minimales complejos son precisamente los triángulos, los cuadriláteros y una clase especial de hexágonos que llamamos hexágonos de Pappus. Usando esto podemos ver que los 3-politopos psd-minimales complejos deben tener vértices de grado 3, 4 o 6 y facetas con 3, 4 o 6 lados...Doctor en MatemáticasDoctorado87 hojasapplication/pdfengUniversidad de los AndesDoctorado en MatemáticasFacultad de CienciasDepartamento de MatemáticasThe slack model in the study of polytopesTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesishttp://purl.org/coar/resource_type/c_db06http://purl.org/coar/version/c_970fb48d4fbd8a85Texthttp://purl.org/redcol/resource_type/TDPolitoposGeometría algebraicaMatemáticasMatemáticas200512792PublicationORIGINAL23522.pdfapplication/pdf603397https://repositorio.uniandes.edu.co/bitstreams/4c8016f6-2f10-4b92-8475-eaddf2418c02/download9045ba130c64137506789160325ceb0eMD51THUMBNAIL23522.pdf.jpg23522.pdf.jpgIM Thumbnailimage/jpeg5427https://repositorio.uniandes.edu.co/bitstreams/3e0f7cb0-4481-41dd-9800-1b5583dfbe7e/download06a2a654a766ba720fb34b37d711ff19MD55TEXT23522.pdf.txt23522.pdf.txtExtracted texttext/plain178250https://repositorio.uniandes.edu.co/bitstreams/b6ad9b1f-4ab7-4757-add3-b81ec7eb2382/download10604af5f155b0026305e18b8016cefcMD541992/51154oai:repositorio.uniandes.edu.co:1992/511542024-08-26 15:23:53.935http://creativecommons.org/licenses/by-nc-sa/4.0/open.accesshttps://repositorio.uniandes.edu.coRepositorio institucional Sénecaadminrepositorio@uniandes.edu.co
dc.title.spa.fl_str_mv The slack model in the study of polytopes
title The slack model in the study of polytopes
spellingShingle The slack model in the study of polytopes
Politopos
Geometría algebraica
Matemáticas
Matemáticas
title_short The slack model in the study of polytopes
title_full The slack model in the study of polytopes
title_fullStr The slack model in the study of polytopes
title_full_unstemmed The slack model in the study of polytopes
title_sort The slack model in the study of polytopes
dc.creator.fl_str_mv Torres Chaves, Juan Camilo
dc.contributor.advisor.none.fl_str_mv Bogart, Tristram
Gouveia, Joáo
dc.contributor.author.none.fl_str_mv Torres Chaves, Juan Camilo
dc.contributor.jury.none.fl_str_mv Padrol, Arnau
Schurmann, Achill
Velasco Gregory, Mauricio Fernando
dc.subject.armarc.none.fl_str_mv Politopos
Geometría algebraica
Matemáticas
topic Politopos
Geometría algebraica
Matemáticas
Matemáticas
dc.subject.themes.none.fl_str_mv Matemáticas
description In this thesis we present some of the applications of studying polytopes via their slack matrices and slack ideals. We prove that McMullen's operations on polytopes, which includes the join, the vertex sum, the vertex splitting and their dual operations, preserve graphicality in the same way it was known it preserves projective uniqueness. We use this result to identify a large class of projectively unique order polytopes; namely, we prove that every ranked finite poset with no 3-antichain has a graphic and thus projectively unique order polytope. We give a complete characterization of complex psd-minimal polygons; we prove that the complex psd-minimal polygons are precisely the triangles, the quadrialterals and a special class of hexagons which we call Pappus' hexagons. Using this, it can be seen that complex psd-minimal 3-polytopes must have vertices of degree 3, 4 or 6 and facets with 3, 4 or 6 sides. We identify all the combinatorial classes of 3-polytopes...
publishDate 2020
dc.date.issued.none.fl_str_mv 2020
dc.date.accessioned.none.fl_str_mv 2021-08-10T18:13:02Z
dc.date.available.none.fl_str_mv 2021-08-10T18:13:02Z
dc.type.spa.fl_str_mv Trabajo de grado - Doctorado
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dc.identifier.pdf.none.fl_str_mv 23522.pdf
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23522.pdf
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language eng
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dc.format.extent.none.fl_str_mv 87 hojas
dc.format.mimetype.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Universidad de los Andes
dc.publisher.program.none.fl_str_mv Doctorado en Matemáticas
dc.publisher.faculty.none.fl_str_mv Facultad de Ciencias
dc.publisher.department.none.fl_str_mv Departamento de Matemáticas
publisher.none.fl_str_mv Universidad de los Andes
institution Universidad de los Andes
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