Exceptional surgeries on knots with exceptional classes

Abstract

We survey 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 β 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, ∂ M), then, with only a few possible exceptions, every 3-manifold M ′ obtained by Dehn surgery on β with surgery distance ∆ ≥ 2 will be hyperbolic.