Crossing change and exceptional Dehn surgery

Abstract

Thurston’s hyperbolic Dehn surgery theorem [11], [12] asserts that if a knot in the 3-sphere 3 is hyperbolic (i.e., 3 − admits a complete hyperbolic structure of finite volume), then all but finitely many Dehn surgeries on yield hyperbolic 3-manifolds. By an exceptional surgery on a hyperbolic knot we mean a nontrivial Dehn surgery producing a non-hyperbolic manifold. Refer to [3], [6] for a survey on Dehn surgery on knots. We empirically know that ‘most’ knots are hyperbolic and ‘most’ hyperbolic knots have no exceptional surgeries. In this paper, we demonstrate the abundance of hyperbolic knots with no exceptional surgeries by showing that every knot is ‘close’ to infinitely many such hyperbolic knots in terms of crossing change. We regard that two knots are the same if they are isotopic in 3. For a knot in 3, let ( ) be the set of knots each of which is obtained by changing at most crossings in a diagram of .