Taut foliations, braid positivity, and unknot detection

TL;DR

This paper investigates positive braid knots and their relation to the L-space conjecture, providing new constructions of taut foliations, a braid positivity obstruction for cable knots, and exploring limitations of these methods.

Contribution

It offers the first construction of taut foliations in certain Dehn surgeries on positive braid knots and introduces a new braid positivity obstruction for cable knots.

Findings

Constructed taut foliations for manifolds with surgery slope r<g(K)+1.

Proved that the (n,±1)-cable of a knot is braid positive only if the knot is the unknot.

Presented examples showing limitations and negative evidence for the L-space conjecture.

Abstract

We study positive braid knots (the knots in the three-sphere realized as positive braid closures) through the lens of the L-space conjecture. This conjecture predicts that if $K$ is a non-trivial positive braid knot, then for all $r < 2g(K)-1$, the 3-manifold obtained via $r$-framed Dehn surgery along $K$ admits a taut foliation. Our main result provides some positive evidence towards this conjecture: we construct taut foliations in such manifolds whenever $r<g(K)+1$. As an application, we produce a novel braid positivity obstruction for cable knots by proving that the $(n,\pm 1)$-cable of a knot $K$ is braid positive if and only if $K$ is the unknot. We also present some curious examples demonstrating the limitations of our construction; these examples can also be viewed as providing some negative evidence towards the L-space conjecture. Finally, we apply our main result to produce taut foliations in some splicings of knot exteriors.