Exceptional surgeries on knots with exceptional classes

Abstract

We survey the aspects of classical combinatorial sutured manifold theory and show how they can be adapted to study exceptional Dehn fillings and 2-handle additions. As a consequence, we show that if a hyperbolic knot \(\beta \) in a compact, orientable, hyperbolic 3-manifold, \(M\) has the property that winding number and wrapping number are not equal with respect to a non-trivial class in \(H_2(M,\partial M)\), and then, with only a few possible exceptions, every 3-manifold \(M'\) obtained by Dehn surgery on \(\beta \) with surgery distance \(\Delta \ge 2\) will be hyperbolic.