Representing links in 3-manifolds by branched coverings of \(S^3\)
The authors introduce a combinatorial representation by means of planar colored diagrams of links in closed 3-manifolds described as branched coverings of the 3-sphere. Then they prove an equivalence theorem based on local moves on the diagrams, and obtain a universal configuration for such a representation. As a consequence, they give unified proofs of two different results on existence of fibered links in 3-manifolds, and finally extend two results obtained respectively by \textit{J. Harer} [Math. Proc. Camb. Philos. Soc. 92, 133-138 (1982; Zbl 0501.57004)] for knots and by \textit{J. R. Stallings} [Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. pure Math., Vol. 32, Part 2, 55-60 (1978; Zbl 0394.57007)] for links in the 3-sphere.
colored diagram representation of a link
scientific article; zbMATH DE number 1215546