The Heegaard genus of manifolds obtained by surgery on links and knots

Abstract

LetL⊂S3be a fixed link. It is shown that there exists an upper bound on the Heegaard genus of any manifold obtained by surgery onL. The tunnel number ofL,T(L), is defined and used as an upper bound. IfK′is a double of the knotK, it is shown thatT(K′)≤T(K)+1. IfMis a manifold obtained by surgery on a cable link aboutKwhich hasncomponents, it is shown that the Heegaard genus ofMis at mostT(K)+n+1.