A cabling conjecture for knots in lens spaces
type: Item
instance of: scholarly article
title: A cabling conjecture for knots in lens spaces
zbMATH Open document ID: 1320.57006
author: Kenneth L. Baker
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2015-01-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1306.0596
review text: The cabling conjecture of González-Acuña-Short [\textit{F. González-Acuña}and \textit{H. Short}, Math. Proc. Camb. Philos. Soc. 99, 89--102 (1986; Zbl 0591.57002)] states that if a knot in \(S^3\) admits a reducible surgery manifold, then the knot is either a torus knot or a cable knot. A strong evidence for it is \textit{J. E. Greene's} result [J. Differ. Geom. 100, No. 3, 491--506 (2015; Zbl 1325.57016)] in the case the surgery manifold is a connected sum of lens spaces. A straightforward generalization of the cabling conjecture from knots in \(S^3\) to knots in lens spaces is false due to an example given by Boyer-Zhang. Using Morton's classification of non-prime links with braid index 3 and Montesino's Trick, the author constructs certain types of bandings to 2-bridge links, and from these many examples of hyperbolic knots in lens spaces admitting non-prime surgery manifolds. It is proved that for each pair of integers \((r,s)\) and slope \(p/q\), there is a knot \(K_{r,s}^{p,q}\) in \(L(r,1) \sharp L(s,1)\) admitting an integral surgery yielding the lens space \(L(u,v)\) (where \(u, v\) are expressed in terms of \(r, s, p, q\)), and if \((r,s) \neq (0,0)\) then infinitely many of these knots are hyperbolic. It follows that in the case say \(s= \pm1\), the knots are in \(L(r,1)\) and have non-trivial lens space surgeries. Considering the surgery dual of the above result, it follows as a corollary that for \((r,s)\) with \(|r|, |s| >1\), there are infinitely many lens spaces containing hyperbolic knots that admit a surgery yielding \(L(r,1) \sharp L(s,1)\). These knots include the previously known examples of hyperbolic knots in non-prime manifolds with lens space surgeries. This leads the author to reformulate a generalization of the Lens Space Cabling Conjecture, as follows. Assume \(K\) is a knot in a lens space \(L\) that has irreducible exterior and admits a surgery to a non-prime 3-manifold. If \(K\) is hyperbolic, then the surgery manifold is \(L(r,1) \sha…
reviewed by: Michael C. Tsau
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 6384873
zbMATH Keywords: cabling conjecture
zbMATH Keywords: reducible manifolds
zbMATH Keywords: Dehn surgery
zbMATH Keywords: 3-braids
zbMATH Keywords: two-bridge links
zbMATH Keywords: lens spaces
Wikidata QID: Q123017444
describes a project that uses: SnapPy
MaRDI profile type: Publication
OpenAlex ID: W2117199360
arXiv ID: 1306.0596
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DOI: 10.1007/S40590-014-0030-X
description: scientific article