A cabling conjecture for knots in lens spaces

Abstract

Closed $$3$$ -string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces $$L(r,1) \# L(s,1)$$ that admit a surgery to a lens space for all pairs of integers $$(r,s)$$ except $$(0,0)$$ . These knots are typically hyperbolic. We also demonstrate that the previously known two families of examples of hyperbolic knots in non-prime manifolds with lens space surgeries of Eudave-Munoz–Wu and Kang all fit this construction. As such, we propose a generalization of the cabling conjecture of Gonzalez-Acuna–Short for knots in lens spaces.