2016/02/19 by Celeste Damiani, Céleste Damiani, Damiani, Celeste +2
Mathematics · #57M25 #57M27 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:57M25 #msc:57M27
paper · pdf · doi:10.48550/arxiv.1602.06191
arxiv created 2016/02/19 · openalex publication_date 2016/02/19 · arxiv updated 2016/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Ribbon tangles are proper embeddings of tori and cylinders in the 4-ball~B4, "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 RibG 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 CobG over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones, and prove that the invariant 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 A and of the algebra CobG, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald. 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.