Communications in Analysis and Geometry

Volume 30 (2022)

Number 10

Jointly primitive knots and surgeries between lens spaces

Pages: 2157 – 2229



Kenneth L. Baker (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)

Neil R. Hoffman (Department of Mathematics, Oklahoma State University, Stillwater, Ok., U.S.A.)

Joan E. Licata (Mathematical Sciences Institute, The Australian National University, Canberra, ACT, Australia)


This paper describes a Dehn surgery approach to generating asymmetric hyperbolic manifolds with two distinct lens space fillings. Such manifolds were first identified in [$\href{}{20}$] 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 $S^3$, 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 [$\href{}{20}$].

Received 6 November 2019

Accepted 1 July 2020

Published 29 September 2023