Jointly primitive knots and surgeries between lens spaces

This paper describes a Dehn surgery approach to generating asymmetric hyperbolic manifolds with two distinct lens space fillings. Such manifolds were first identified in work of Dunfield-Hoffman-Licata as the result of a computer search of the SnapPy census, but the current work establishes a topological framework for constructing vastly many more such examples. We introduce the notion of a jointly primitive'' presentation of a knot and show that a refined version of this condition longitudinally jointly primitive’’ is equivalent to being surgery dual to a $(1,2)$–knot in a lens space. This generalizes Berge’s equivalence between having a doubly primitive presentation and being surgery dual to a $(1,1)$–knot in a lens space. Through surgery descriptions on a seven-component link in $S^3$, we provide several explicit multi-parameter infinite families of knots in lens spaces with longitudinal jointly primitive presentations and observe among them all the examples previously seen in Dunfield-Hoffman-Licata.