Jointly primitive knots and surgeries between lens spaces

TL;DR

This paper develops a topological framework for constructing many asymmetric hyperbolic manifolds with two lens space fillings using a Dehn surgery approach, generalizing previous classifications and providing explicit infinite families.

Contribution

It introduces the notion of 'jointly primitive' and 'longitudinally jointly primitive' presentations, linking them to surgeries on lens spaces and expanding the known examples.

Findings

Established a topological construction method for asymmetric hyperbolic manifolds with lens space fillings.

Connected jointly primitive presentations to surgery duals of (1,2)-knots in lens spaces.

Provided explicit multi-parameter infinite families of such knots in lens spaces.

Abstract

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.