Dehn fillings of knot manifolds containing essential once-punctured tori (Q2862133)

The paper under review is set as the second of four papers planned by the authors toward a conjecture raised by the second author. The conjecture concerns exceptional fillings (i.e., Dehn fillings yielding non-hyperbolic manifolds) on hyperbolic knot manifolds (i.e., compact, connected, orientable, hyperbolic 3-manifold whose boundary is a torus). In particular, it is conjectured that only four specific hyperbolic knot manifolds can admit pairs of exceptional fillings with distance greater than 5. It is already known that the maximal distance between exceptional fillings is 8 by \textit{M. Lackenby} and \textit{R. Meyerhoff} [Invent. Math. 191, No. 2, 341--382 (2013; Zbl 1263.57013)], and there are infinitely many such manifolds admitting exceptional fillings with distance 5. Recall that exceptional fillings are classified into reducible, toroidal, or Seifert fibered. Then actually the conjecture is now known to be true unless one of the fillings gives a small Seifert fibered manifold, that is, a Seifert fibered manifold with three singular fibers over the 2-sphere.NEWLINENEWLINE In the series of papers, the authors consider the case of a pair of exceptional fillings, one of which is small Seifert fibered and the other is toroidal. In this case, the conjecture above is reduced to: If a hyperbolic knot manifold admits a small Seifert fibered filling and a toroidal filling of distance greater than 5, then the manifold is the figure-eight knot exterior. In the first of the series of papers [Adv. Math. 230, No. 4--6, 1673--1737 (2012; Zbl 1248.57004)], they showed that if a hyperbolic knot manifold admits a pair of small Seifert fillings and toroidal filling along the slope is not the boundary slope of an essential punctured torus which is a fibre or semi-fibre, or which has fewer than three boundary components, then the distance of the pair of fillings is at most 5. Then, in the current paper under review, they focus on the case where the punctured torus has only one boundary component, and prove that the conjecture above is also true in this case. Precisely they show that the distance of a small Seifert filling and the filling along the boundary slope of an essential once-punctured torus is at most 5. Furthermore, they obtain a complete classification of the manifolds and such pairs of fillings with distance 4 and 5.