L-space knot conjecture (characterization of persistently foliar knots)

Determine whether a knot K in S^3 is persistently foliar if and only if K is not an L-space knot and K has no reducible surgeries. Here, persistently foliar means that for each non-meridional slope on K there exists a coorientable taut foliation in the exterior of K intersecting the boundary torus transversely in parallel curves of that slope, and an L-space knot is a knot admitting some positive surgery that yields an L-space.

Background

The L-space knot conjecture, proposed by Delman and Roberts, refines the L-space conjecture to the setting of all surgeries on a fixed knot. It predicts a sharp dichotomy: knots with any L-space surgery (or reducible surgery) fail to be persistently foliar, while all others should be persistently foliar.

This paper provides sufficient diagrammatic conditions ensuring a knot is persistently foliar, thus verifying the conjectured behavior for broad classes including many arborescent knots and certain braid closures. It also develops tools that adapt to links, yielding complete (1)⇔(2) equivalence for surgeries on the Borromean link.