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Alexander invariants of ribbon tangles and planar algebras

Abstract : 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 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 Rib(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 Cob(G) over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones [Jon99], and prove that the invariant 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 A and of the algebra Cob(G), and observe that our construction is a topological incarnation of the Alexander invariant of Archibald [Arc10]. When restricted to diagrams without virtual crossings, 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].
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Submitted on : Tuesday, June 4, 2019 - 4:23:54 PM
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Celeste Damiani, Vincent Florens. Alexander invariants of ribbon tangles and planar algebras. Journal of the Mathematical Society of Japan, Maruzen Company Ltd, 2018, 70 (3), pp.1063-1084. ⟨10.2969/jmsj/75267526⟩. ⟨hal-02147424⟩

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