A cabling conjecture for knots in lens spaces
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 \(L(r,1) \sharp L(s,1)\), and for some slope \(p/q\) the surgery dual is the knot \(K_{r,s}^{p,q}\) in the lens space stated in the main theorem. Otherwise either \(K\) is a torus knot embedded in the Heegaard torus, or a knot embedded in a Klein bottle with annular complement, or a cable knot; and the surgery is along the boundary slope of an essential annulus in the exterior of \(K\). It follows from a theorem of Boyer-Zhang that non-hyperbolic knots in lens spaces satisfy the Lens Space Cabling Conjecture proposed by the author.
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