Generalization of Artin's theorem on the isotopy of closed braids. I (Q6610182)

In this work, the author considers isotopies of transversal links in an arbitrary compact 3-manifold which is fibered over a circle with compact fibers. In particular, he shows that in the case when \(M\) is compact, orientable, and locally fibered over the circle with compact fiber, two transverse links \(L_1\) and \(L_2\) are ambient isotopic in \(M\) if and only if they are transversely isotopic. Stated differently, in this setting, it suffices to determine if \(L_1\) and \(L_2\) are ambient isotopic in \(M\) to decide if there exists a transverse isotopy between them. This is a generalization of the fact that two closed braids in a solid torus are ambient isotopic if and only if they represent the same conjugacy class in the braid group (are transversely isotopic).\N\NThe main result is obtained in two steps:\N\N1) Showing that two PL transversal links that are topologically isotopic as transversal links are, in fact, PL ambiently isotopic. This amounts to showing locally these two types of isotopies are equivalent then showing this local equivalence implies a global equivalence.\N\N2) Showing two PL ambient isotopic transversal PL links are topologically isotopic as transverse links. This was shown by using results about isotopies of incompressible surfaces in fibered spaces.\N\NFor related work on transverse links see [\textit{S. Yu. Orevkov} and \textit{V. V. Shevchishin}, J. Knot Theory Ramifications 12, No. 7, 905--913 (2003; Zbl 1046.57007)].