  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 $\mathsf{A}$ 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 $\mathsf{R}ib_G$ 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 $\mathsf{C}ob_G$ over
the operad of smooth cobordisms, inspired by diagrammatic planar algebras
introduced by Jones, and prove that the invariant $\mathsf{A}$ commutes with
the compositions in this algebra. On the other hand, ribbon tangles admit
diagrammatic representations, throught welded diagrams. We give a simple
combinatorial description of $\mathsf{A}$ and of the algebra $\mathsf{C}ob_G$,
and observe that our construction is a topological incarnation of the Alexander
invariant of Archibald. When restricted to diagrams without virtual crossings,
$\mathsf{A}$ provides a purely local description of the usual Alexander
poynomial of links, and extends the construction by Bigelow, Cattabriga and the
second author.

Geometric Topology