Exceptional surgeries on knots with exceptional classes

TL;DR

This paper explores how classical sutured manifold theory can be adapted to study exceptional Dehn fillings on hyperbolic knots, showing that most surgeries with certain properties preserve hyperbolicity.

Contribution

It introduces a novel adaptation of sutured manifold theory to analyze exceptional surgeries on knots, providing new conditions for hyperbolicity preservation.

Findings

Most Dehn surgeries with distance ≥ 2 on certain hyperbolic knots yield hyperbolic manifolds.

Winding and wrapping number discrepancies influence the hyperbolic nature of resulting manifolds.

The approach narrows down exceptions where surgeries do not preserve hyperbolicity.

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 $M'$ obtained by Dehn surgery on $\beta$ with surgery distance $\Delta \geq 2$ will be hyperbolic.