Non degenerating Dehn fillings on genus two Heegaard splittings of knots' complements
type: Item
instance of: scholarly article
title: Non degenerating Dehn fillings on genus two Heegaard splittings of knots' complements
zbMATH Open document ID: 1396.57012
author: Jiming Ma
author: Ruifeng Qiu
author: Yanqing Zou
published in: Science China. Mathematics
publication date: 2018-07-05T00:00:00Z
review text: 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 the…
reviewed by: Kun Du
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6898604
zbMATH Keywords: Heegaard distance
zbMATH Keywords: Dehn filling
zbMATH Keywords: hyperbolic knot
zbMATH Keywords: curve complex
MaRDI profile type: Publication
full work available at URL: s11425-017-9093-5
OpenAlex ID: W2765213804
cites work: Bounds on exceptional Dehn filling. II
cites work: Q3910741
cites work: 3-manifolds as viewed from the curve complex
cites work: Heegaard splittings of distance exactly \(n\)
cites work: The maximal number of exceptional Dehn surgeries
cites work: Heegaard surfaces and measured laminations. I: the Waldhausen conjecture
cites work: Images of the disk complex
cites work: Distance degenerating handle additions
cites work: Geometry of the complex of curves. I: Hyperbolicity
cites work: Geometry of the complex of curves. II: Hierarchical structure
cites work: The geometry of the disk complex
cites work: High distance knots
cites work: The Heegaard distances cover all nonnegative integers
cites work: Proximity in the curve complex: boundary reduction and bicompressible surfaces
cites work: Heegaard-Zerlegungen der 3-Sphäre
cites work: Infinitely many hyperbolic 3-manifolds admitting distance-\(d\) and genus-\(g\) Heegaard splittings
cites work: Unstabilized self-amalgamation of a Heegaard splitting
cites work: Q5498118
DOI: 10.1007/S11425-017-9093-5
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description: scientific article; zbMATH DE number 6898604