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 $r<2g\left(K\right)-1$, 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 $r<g\left(K\right)+1$. 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.