Generalization of Artin’s Theorem on the Isotopy of Closed Braids. I

A classical theorem of braid theory, dating back to Artin’s work,says thattwo closed braids in a solid torus are ambient isotopicif and only ifthey represent the same conjugacy class of the braid group.This theorem can be reformulatedin the framework of link theorywithout referring to the group structure.A link in a surface bundle over the circle is transversalwhenever it covers the circle.In this terminology,Artin’s theorem states thatin a solid torus trivially fibered over the circletransversal links are ambient isotopicif and only ifthey are isotopic in the class of transversal links.We generalize this result by proving that(in the piecewise linear category)transversal links in an arbitrary compact orientable $ 3 $ -manifoldfibered over the circle with a compact fiberare ambient isotopicif and only ifthey are isotopic in the class of transversal links.