Abstract
Closed
$$3$$
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
$$L(r,1) \# L(s,1)$$
L
(
r
,
1
)
#
L
(
s
,
1
)
that admit a surgery to a lens space for all pairs of integers
$$(r,s)$$
(
r
,
s
)
except
$$(0,0)$$
(
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-Muñoz–Wu and Kang all fit this construction. As such, we propose a generalization of the cabling conjecture of González-Acuña–Short for knots in lens spaces.
