Taut foliations in knot complements

TL;DR

This paper proves that for any nontrivial knot in the 3-sphere, there exists an interval of Dehn surgery slopes near zero that produce 3-manifolds with taut foliations, extending Gabai's zero frame surgery result.

Contribution

It generalizes Gabai's theorem by showing a continuous range of slopes near zero yield taut foliations in knot complements.

Findings

Existence of an open interval of slopes near zero producing taut foliations.

Extension of Gabai's theorem to a range of Dehn surgeries.

Applicable to all nontrivial knots in $S^3$.

Abstract

We show that for any nontrivial knot in $S^3$, there is an open interval containing zero such that a Dehn surgery on any slope in this interval yields a 3-manifold with taut foliations. This generalizes a theorem of Gabai on zero frame surgery.