Abstract: We draw some parallels between results describing the homology group of branched cyclic coverings over knots and results in the theory of cyclic coverings over graphs. We present a correspondence between objects of knot theory and their analogues in graph theory: ∙a knot in the sphere ^3 corresponds to a vertex and of the cone { }⋆ ; ∙the Alexander polynomial of corresponds to the associated Laurent polynomial of the graph ; ∙the complement ^3 ∖ corresponds to the graph ; ∙a cyclic covering over ^3 ∖ corresponds to a cyclic covering over the graph ; ∙the cyclic covering _ of the sphere ^3 branched over corresponds to the cyclic covering _ of the cone { }⋆ branched over ; ∙the homology group _1( _ , ) corresponds to the Jacobian group ( _ ).

Cite: Mednykh A.
Homology group of branched coverings of knots and Jacobians of graphs
International Conference and PhD-Master Summer School "Groups and Graphs, Algebras and Applications" 04-17 Aug 2025