Equivalence of a bridged link calculus and Kirbyʹs calculus of links on nonsimply connected 3-manifolds

We recall an extension of Kirbyʹs calculus on nonsimply connected 3-manifolds given by Fenn and Rourke (1979), and the surgery calculus of bridged links from Kerler (1994), which involves only local moves. We give a short combinatorial proof that the two calculi are equivalent, and thus describe the same classes of 3-manifolds. This makes the proofs for the validity of surgery calculi of Fenn and Rourke and Kerler interchangeable.