Crossing change and exceptional Dehn surgery (Q1860762)

The authors investigate the problem how the hyperbolic structure of a knot (more precisely, of its complement in \(S^3\)) can vary while changing some crossings of the knot. The authors determine a class of knots \(B_{n}(K)\) which can be obtained from \(K\) by changing at most n crossings. The main theorem of the work states the following. For every knot \(K\) in \(S^3\), the set \(B_{1}(K)\) contains infinitely many hyperbolic knots, each of which has no exceptional surgeries. Any arbitrary knot can be deformed into a hyperbolic knot with no exceptional surgeries by only one crossing change. The authors also state some open questions concerning prime knots and hyperbolic knots.