Taut Foliations, Positive 3-Braids, and the L-Space Conjecture

TL;DR

This paper constructs taut foliations in certain 3-manifolds obtained by Dehn surgery on positive 3-braid knots, confirming predictions of the L-space Conjecture and providing new examples for hyperbolic L-space knots.

Contribution

It provides the first construction of taut foliations for all non-L-space manifolds obtained from surgeries on an infinite family of hyperbolic L-space knots.

Findings

Taut foliations exist in all non-L-spaces from surgeries on specific positive 3-braid knots.

Constructs taut foliations in manifolds from surgeries on positive 1-bridge braids.

Confirms the L-space Conjecture predictions for these classes of knots.

Abstract

We construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 3-braid knot $K$ in $S^3$, where $r < 2g(K)-1$ and $g(K)$ denotes the Seifert genus of $K$. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obtained by surgery along the pretzel knot $P(-2,3,7)$, and indeed along every pretzel knot $P(-2,3,q)$, for $q$ a positive odd integer. This is the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. Additionally, we construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 1-bridge braid in $S^3$, where $r <g(K)$.