Non degenerating Dehn fillings on genus two Heegaard splittings of knots' complements
Let \(M\) be a hyperbolic \(3\)-manifold such that \(\partial M\) is a torus, \(M(r)\) be a Dehn filling of \(M\) along a slope \(r\) on \(\partial M\). By Thurston's Theorem, there are at most finitely many exceptional slopes \(r\) such that \(M(r)\) is non-hyperbolic. \textit{M. Scharlemann} and \textit{Y.-Q. Wu} [J. Aust. Math. Soc., Ser. A 55, No. 1, 72--89 (1993; Zbl 0802.57005)] considered a \(3\)-manifold with higher genus boundary \(F_{1}\) and showed that the set of basic degenerating curves on \(F_{1}\) is finite and bounded by a universal constant only depending on \(g(F_{1})\). \textit{M. Lackenby} [Geom. Topol. 6, 889--904 (2002; Zbl 1021.57010)] considered a \(3\)-manifold \(M\) which has a finite volume complete hyperbolic metric with totally geodesic boundary and possibly some cusps and showed that there is a finite set \(\mathcal{C}\) of simple closed curves on \(\partial M\) such that if none of the curves in \(\mathcal{C}\) bounds a disk in the attached handlebodies \(H\) then the resulting manifold is irreducible, atoroidal, word hyperbolic and not Seifert fibered. It is also shown that the number of curves one needs to exclude from each boundary component of \(M\) is bounded by a constant depending only on the genus of that component. \textit{L. Liang} et al. [Proc. Am. Math. Soc. 144, No. 1, 423--434 (2016; Zbl 1327.57022)] considered the strongly irreducible Heegaard splitting \(M=V\cup_{S}W\) and showed that if \(V\) is simple or \(M=V\cup_{S}W\) is locally complicated, then the diameter of the set of distance degenerating curves on \(F_{1}\) is bounded by a constant depending only on \(g(S)\). In the paper under review, the authors consider Dehn fillings on genus two Heegaard splittings of the complement \(E(K)\) of a knot \(K\) in \(S^{3}\) and prove that if the distance of a given genus two Heegaard splitting of \(E(K)\) is at least \(3\), then for any two distance degenerating slopes on \(\partial E(K)\), there is a universal bound of their distance in the curve complex of \(\partial E(K)\).
scientific article; zbMATH DE number 6898604
scientific article