Reducible surgeries and Heegaard Floer homology (Q2355769)

The cabling conjecture claims that if a knot in the \(3\)-sphere admits Dehn surgery yielding a reducible manifold then the knot is cabled and the slope is given by the cabling annulus. This conjecture has been solved affirmatively for torus knots and satellite knots, so the remaining case is for hyperbolic knots. In other words, any hyperbolic knot is expected to admit no reducible surgery. For any reducible surgery, it is known that the slope is integral. Any knot admits at most two reducible surgeries, and if there are two, then these correspond to successive integers. Moreover, \textit{D. Matignon} and \textit{N. Sayari} [Hiroshima Math. J. 33, No. 1, 127--136 (2003; Zbl 1029.57007)] showed that for a non-cable knot, any reducing slope \(p\) lies in the range \(1<p\leq 2g-1\), where \(g\) denotes the genus of the knot. The main result of the paper under review claims that if a hyperbolic knot admits a positive \(L\)-space surgery, then any reducing slope for the knot is equal to \(2g-1\). An \(L\)-space is a rational homology sphere with the simplest Heegaard Floer homology. If a knot admits an \(L\)-space surgery, then the knot or its mirror image admits such a positive surgery. As said above, the cabling conjecture claims that any hyperbolic knot has no reducible surgery. Hence the main result gives a strong constraint for a reducing slope when a hyperbolic knot has an \(L\)-space surgery. The arguments are based on Heegaard Floer homology theory. As a corollary, we have that any \(L\)-space knot cannot admit two reducible surgeries. Genus one knots are known to have no reducible surgery, but the authors propose a new proof and get the same answer for genus two knots.