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Ribbon tangles are proper embeddings of tori and cylinders in the$4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks assingularities. We construct an Alexander invariant $\mathsf{A}$ of ribbontangles equipped with a representation of the fundamental group of theirexterior in a free abelian group $G$. This invariant induces a functor in acertain category $\mathsf{R}ib_G$ of tangles, which restricts to the exteriorpowers of Burau-Gassner representation for ribbon braids, that are analogous tousual braids in this context. We define a circuit algebra $\mathsf{C}ob_G$ overthe operad of smooth cobordisms, inspired by diagrammatic planar algebrasintroduced by Jones, and prove that the invariant $\mathsf{A}$ commutes withthe compositions in this algebra. On the other hand, ribbon tangles admitdiagrammatic representations, throught welded diagrams. We give a simplecombinatorial description of $\mathsf{A}$ and of the algebra $\mathsf{C}ob_G$,and observe that our construction is a topological incarnation of the Alexanderinvariant of Archibald. When restricted to diagrams without virtual crossings,$\mathsf{A}$ provides a purely local description of the usual Alexanderpoynomial of links, and extends the construction by Bigelow, Cattabriga and thesecond author.

Alexander invariants of ribbon tangles and planar algebras