High distance knots (Q2464801)

In the paper under review, for any pair of integers \(g\) and \(n\), the authors present a knot in the 3-sphere with the exterior admitting a Heegaard splitting of genus \(g\) having distance greater than \(n\). The distance of a Heegaard splitting, which is an invariant reflecting well the complexity of a 3-manifold, was defined by \textit{J. Hempel} [Topology 40, No. 3, 631--637 (2001; Zbl 0985.57014)]. Roughly speaking it is the distance between the two disk complexes defined by \textit{D. McCullough} [J. Differ. Geom. 33, No.~1, 1--65 (1991; Zbl 0721.57008)] for compression bodies bounded by a Heegaard surface. Here we regard them as subcomplexes in the curve complex of the Heegaard surface. In that paper, Hempel gave examples of 3-manifolds with arbitrary high distance Heegaard splitting, and from these, it is easy to find knots with exteriors having arbitrary high distance Heegaard splitting in some 3-manifolds. However, to construct such a knot in the prescribed ambient manifold is a challenging problem, which is affirmatively answered in the paper under review for the case that the manifold is the 3-sphere. One of the key ingredients of the authors' proof is to find a pseudo-Anosov gluing map for a Heegaard splitting of some knot exterior. Also, as a corollary, they give a knot in the 3-sphere with tunnel number \(t\) so that the knot has no \((t,b)\)-decomposition for any pair of integers \(t\) and \(b\).