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Homology spheres yielding lens spaces. (English) Zbl 1441.57016

Akbulut, Selman (ed.) et al., Proceedings of the 24th Gökova geometry-topology conference, Gökova, Turkey, May 29 –June 2, 2017. Somerville, MA: International Press; Gökova: Gökova Geometry-Topology Conferences (GGT). 73-121 (2018).
Let \(K\) be a knot in a homology 3-sphere \(Y\); \(K\) is called a lens space knot if some integral surgery on \(K\) produces a lens space. J. Berge in an unpublished manuscript constructed 7 families of lens space knots in the 3-sphere. These knots come from a double primitive construction, which can be defined more in general, not just for \(S^3\). Let \(K\) be a knot in a homology 3-sphere \(Y\) of Heegaard genus at most \(2\), and let \(Y=H_1 \cup_{\Sigma} H_2\) be a Heegaard decomposition of \(Y\) of genus two, with Heegaard surface \(\Sigma\). The knot \(K\) is a double-primitive knot if it can be isotoped to lie in \(\Sigma\) in such a way that \(K\) is a primitive curve in each handlebody \(H_i\). Then Dehn surgery on \(K\) with the slope given by the intersection of a regular neighboorhod of \(K\) with \(\Sigma\) produces a lens space.
In a previous paper [Exp. Math. 18, No. 3, 285–301 (2009; Zbl 1187.57011)], the present author gave 20 families of lens spaces knots in the Poincaré homology sphere \(\Sigma(2,3,5)\), which are double-primitive knots, and such that the dual knots (that is, the core of the surgered solid torus) are certain \((1,1)\)-knots in the corresponding lens spaces. In this paper, concrete diagrams of the 20 families of knots are given. This is extended to other Brieskorn homology spheres, and tables of lens spaces knots are given for the Brieskorn homology spheres \(\Sigma(2,3,7)\), \(\Sigma(2,3,6n \pm 1)\), \(\Sigma(2,2s+a,2(2s+1) \pm 1)\), \(\Sigma(2,2s+a,2(2s+1)n \pm 1)\). Some examples are also given of double-primitive knots for some graph homology spheres. Some conjectures are posed about lens spaces knots in homology spheres.
For the entire collection see [Zbl 1398.53004].

MSC:

57K30 General topology of 3-manifolds
57R65 Surgery and handlebodies

Citations:

Zbl 1187.57011
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