If we consider the set of manifolds that can be obtained by surgery on
a fixed knot K, then we have an associated set of numbers corresponding to the Heegaard genus of these manifolds. It is known that there is an upper bound to this set of numbers. A knot K is said to have Property R+ if longitudinal surgery yields a manifold of highest possible Heegaard genus among those obtainable by surgery on K. In this paper we show that torus knots, 2-bridge knots, and knots which are the connected sum of arbitrarily many (2,m)-torus knots have Property R+ It is shown that if K is constructed from the tangles (B1,t1),(B2,t2),…,(Bn,tn) then T(K)≤1+∑i=1nT(Bi,ti) where T(K) is the tunnel of K and T(Bi,ti) is the tunnel number of the tangle (Bi,ti). We show that there exist prime knots of arbitrarily high tunnel number that have Property R+ and that manifolds of arbitrarily high Heegaard genus can be obtained by surgery on prime knots.
