Homotopy classification of ribbon tubes and welded string links
| Autor: | Emmanuel Wagner, Jean-Baptiste Meilhan, Benjamin Audoux, Paolo Bellingeri |
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| Přispěvatelé: | Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), ANR-11-JS01-0002,VasKho,De Vassiliev à Khovanov – Invariants de type fini et Categorification pour les objets noués(2011), Institut de Mathématiques de Marseille ( I2M ), Aix Marseille Université ( AMU ) -Ecole Centrale de Marseille ( ECM ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques Nicolas Oresme ( LMNO ), Université de Caen Normandie ( UNICAEN ), Normandie Université ( NU ) -Normandie Université ( NU ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Fourier ( IF ), Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-11-JS01-0002,VasKho,De Vassiliev à Khovanov – Invariants de type fini et Categorification pour les objets noués ( 2011 ), Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS) |
| Rok vydání: | 2014 |
| Předmět: |
[ MATH.MATH-GT ] Mathematics [math]/Geometric Topology [math.GT]
Pure mathematics 57Q45 57M27 20F36 01 natural sciences String (physics) Theoretical Computer Science Mathematics - Geometric Topology Mathematics (miscellaneous) [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] Mathematics::Quantum Algebra Mathematics::Category Theory 0103 physical sciences Ribbon FOS: Mathematics 0101 mathematics Quotient MSC: 57Q45 (primary) 57M25 57M27 57Q35 (secondary) Mathematics Homotopy 010102 general mathematics Geometric Topology (math.GT) Mathematics::Geometric Topology Virtual knot Action (physics) Free group Gravitational singularity 010307 mathematical physics |
| Zdroj: | Annali della Scuola Normale Superiore di Pisa, Classe di Scienze Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore 2017, 17 (2), pp.713-761. ⟨10.2422/2036-2145.201507_003⟩ Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2017, 17 (2), pp.713-761. 〈10.2422/2036-2145.201507_003〉 Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2017, 17 (2), pp.713-761. ⟨10.2422/2036-2145.201507_003⟩ |
| ISSN: | 0391-173X 2036-2145 |
| DOI: | 10.48550/arxiv.1407.0184 |
| Popis: | Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper we consider ribbon tubes and ribbon torus-links, which are natural analogues of string links and links, respectively. We show how ribbon tubes naturally act on the reduced free group, and how this action classifies ribbon tubes up to link-homotopy, that is when allowing each component to cross itself. At the combinatorial level, this provides a classification of welded string links up to self-virtualization. This generalizes a result of Habegger and Lin on usual string links, and the above-mentioned action on the reduced free group can be refined to a general "virtual extension" of Milnor invariants. As an application, we obtain a classification of ribbon torus-links up to link-homotopy. Comment: 33p. ; v2: typos and minor corrections ; v3: Introduction rewritten, exposition revised, references added. Section 5 of the previous version was significantly expanded and was separated into another paper (arXiv:1507.00202) ; v4: typos and minor corrections ; to appear in Annali della scuola Normale Superiore de Pisa (classe de scienze) |
| Databáze: | OpenAIRE |
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