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 [20] 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 S³, 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 [20].