Jointly primitive knots and surgeries between lens spaces (Q6083954)

Let \(Y\) be a closed \(3\)-manifold, and let \(M\) be the exterior of a two-component link \(C_0\cup C_1\) in \(Y\). Suppose that \(M\) contains a properly embedded, twice-punctured torus \(\Sigma\) with handlebody complement \(H\). A framed knot \(K\) in \(M\) has a jointly primitive presentation in \(\Sigma\) if the two impressions \(K_+\) and \(K_-\) in the boundary of the neighborhood of \(\Sigma\) are a jointly primitive pair of curves in \(H\). This is equivalent to the condition that attaching \(2\)-handles to \(H\) along any subset of \(\{K_+,K_-\}\) yields a handlebody. In particular, if \(\partial \Sigma\) meets either \(\partial N(C_0)\) or \(\partial N(C_1)\) in a longitudinal curve, then the presentation is said to be longitudinal. The main result of the present paper shows that a framed knot \(K\) has a longitudinal jointly primitive presentation if and only if its framed surgery dual \(K^*\) is a \((1,2)\)-knot in a lens space. That is, \(K^*\) meets each solid torus of the Heegaard splitting of a lens space in two mutually trivial arcs. If \(\Sigma\) is a fiber in a fibration of \(Y-N(C_0\cup C_1)\), then the jointly primitive presentation is said to be fibered. The above implies that if a framed knot in a closed \(3\)-manifold has a doubly primitive presentation, then it admits a longitudinally jointly primitive presentation. Several explicit multi-parameter infinite families of jointly primitive knots in lens spaces or connected sums of lens spaces are constructed through surgery descriptions with beautiful pictures. The knots are generically hyperbolic and asymmetric shown through a computer calculation on SnapPy. They cover the previously known 22 hyperbolic manifolds with Heegaard genus 3 and a pair of lens space fillings found by \textit{N. Dunfield} et al. [Math. Res. Lett. 22, No. 6, 1679--1698 (2015; Zbl 1351.57022)].