... Bcorresponding to the subgroup of 771(M-B)representedontopermutationswhichfix1,undthencompleting to f:N->Mbyfillingin the tubularneighborhood of B and extendingf' to f.Aregularbranchedcoveringisoneforwhichf':N') M - Bisaregu-larcovering,or inotherwords,oneforwhich the subgroup of 771(M-B)inquestionisnormal.Amongtheseare the cyclicbranchedcoverings,givenbyrepresentations771(M-B)->Znonto the cyclicgroup of ordern,suchthat the projectionf:N ... HI(M;Z),butthisconditionisnotalwaysnecessary.§4. Commuting the twooperationsIfonehasinhandabranchedcoveringspace, and asurgery to beper-formed on the basemanifold,onemayaskwhether the surgerycanbelifted to the coveringmanifoldinsuchawaythat the surgeredmanifoldupstairsnaturallybranchedcovers the surgeredmanifolddownstairs. The answer to thisisveryinteresting,becauseitshowsonehow to change the orderinwhich the twooperationsareperformed,withoutchanging the resulting3-manifold.Letf:N->Mbeann-foldbranchedcovering of the oriented3-manifoldMalongBeMgivenby ... k(Z and SYMMETRICFIBEREDLINKS9neighborhoodsTi of the ci's and regluing the closedneighborhoodsbyorientationpreservingself-homeomorphismscPi:aTi aTi of theirboundary.Itisnothard to seethatNisdeterminedup to homeomorphismby the homologyclasses of the imagecurvescPi(mi)inHI(aTi;Z),wheremiisameridianonaTi(i.e.,miisanorientedsimpleclosedcurveonaTiwhichspansadiskinT...