International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 3, Pages 511-519
Knots with proprety
Department of Mathematics and Statistics, University of Southwestern Louisiana, Lafayette 70504, Louisiana, USA
Received 29 October 1982
Copyright © 1983 Bradd Evans Clark. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
If we consider the set of manifolds that can be obtained by surgery on
a fixed knot , 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 is said to have Property if longitudinal surgery yields a manifold of highest possible Heegaard genus among those obtainable by surgery on . In this paper we show that torus knots, -bridge knots, and knots which are the connected sum of arbitrarily many -torus knots have Property It is shown that if is constructed from the tangles then where is the tunnel of and is the tunnel number of the tangle . We show that there exist prime knots of arbitrarily high tunnel number that have Property and that manifolds of arbitrarily high Heegaard genus can be obtained by surgery on prime knots.