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Let L⊂S3 be a fixed link. It is shown that there exists an upper bound on the Heegaard genus of any manifold obtained by surgery on L. The tunnel number of L, T(L), is defined and used as an upper bound. If K′ is a double of the knot K, it is shown that T(K′)≤T(K)+1. If M is a manifold obtained by surgery on a cable link about K which has n components, it is shown that the Heegaard genus of M is at most T(K)+n+1

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