Taut foliations, braid positivity, and unknot detection

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.

92 pages, 49 figures, 5 tables, 1 flowchart, 1 appendix