Alexander invariants of ribbon tangles and planar algebras
| Autor: | Vincent Florens, Celeste Damiani |
|---|---|
| Přispěvatelé: | Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), School of Mathematics - University of Leeds, University of Leeds, School of Mathematics [Leeds] |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Fundamental group General Mathematics 01 natural sciences Planar algebra Mathematics - Geometric Topology [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] Mathematics::Category Theory Mathematics::Quantum Algebra 0103 physical sciences Ribbon 57M25 57M27 FOS: Mathematics [MATH]Mathematics [math] 0101 mathematics Invariant (mathematics) Mathematics Functor tangles 010102 general mathematics Torus Geometric Topology (math.GT) welded knots 16. Peace & justice Mathematics::Geometric Topology Free abelian group 57M27 57Q45 57M25 Gravitational singularity 010307 mathematical physics Alexander polynomials planar algebras |
| Zdroj: | J. Math. Soc. Japan 70, no. 3 (2018), 1063-1084 Journal of the Mathematical Society of Japan Journal of the Mathematical Society of Japan, Maruzen Company Ltd, 2018, 70 (3), pp.1063-1084. ⟨10.2969/jmsj/75267526⟩ |
| ISSN: | 0025-5645 |
| DOI: | 10.2969/jmsj/75267526⟩ |
| Popis: | Ribbon tangles are proper embeddings of tori and cylinders in the 4-ball $B^4$, “bounding” 3-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathbf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathbf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau–Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathbf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones [Jon99], and prove that the invariant $\mathbf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, through welded diagrams. We give a simple combinatorial description of $\mathbf{A}$ and of the algebra $\mathbf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald [Arc10]. When restricted to diagrams without virtual crossings, $\mathbf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author [BCF15]. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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