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...
- 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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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 |
dc.type.coarversion.fl_str_mv |
http://purl.org/coar/version/c_970fb48d4fbd8a85 |
dc.type.driver.spa.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
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http://purl.org/coar/resource_type/c_db06 |
dc.type.content.spa.fl_str_mv |
Text |
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http://purl.org/redcol/resource_type/TD |
format |
http://purl.org/coar/resource_type/c_db06 |
dc.identifier.uri.none.fl_str_mv |
http://hdl.handle.net/1992/51154 |
dc.identifier.doi.none.fl_str_mv |
10.57784/1992/51154 |
dc.identifier.pdf.none.fl_str_mv |
23522.pdf |
dc.identifier.instname.spa.fl_str_mv |
instname:Universidad de los Andes |
dc.identifier.reponame.spa.fl_str_mv |
reponame:Repositorio Institucional Séneca |
dc.identifier.repourl.spa.fl_str_mv |
repourl:https://repositorio.uniandes.edu.co/ |
url |
http://hdl.handle.net/1992/51154 |
identifier_str_mv |
10.57784/1992/51154 23522.pdf instname:Universidad de los Andes reponame:Repositorio Institucional Séneca repourl:https://repositorio.uniandes.edu.co/ |
dc.language.iso.none.fl_str_mv |
eng |
language |
eng |
dc.rights.uri.*.fl_str_mv |
http://creativecommons.org/licenses/by-nc-sa/4.0/ |
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info:eu-repo/semantics/openAccess |
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http://creativecommons.org/licenses/by-nc-sa/4.0/ http://purl.org/coar/access_right/c_abf2 |
eu_rights_str_mv |
openAccess |
dc.format.extent.none.fl_str_mv |
87 hojas |
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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 |
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Universidad de los Andes |
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Universidad de los Andes |
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