Knots with proprety R +

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 ( B 1 , t 1 ) , ( B 2 , t 2 ) , … , ( B n , t n ) then T ( K ) ≤ 1 + ∑ i = 1 n T ( B i , t i ) where T ( K ) is the tunnel of K and T ( B i , t i ) is the tunnel number of the tangle ( B i , t i ) . 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.