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 $\beta$ 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 $H_2(M,\boundary M)$, then, with only a few possible exceptions, every 3-manifold $M^{\prime }$ obtained by Dehn surgery on $\beta$ with surgery distance $\Delta \ge 2$ will be hyperbolic.

This paper is written in the style of a survey article. Comments are welcome

References in corpus (3)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Band taut sutured manifolds