Once-punctured tori and knots in Lens spaces (Q413401)

This paper addresses the question of when a knot in a lens space has a once-punctured torus properly embedded in its exterior. The first theorem gives a complete list of such knots under the conditions that the lens space is not \(S^1\times S^2\) and the knot is not null-homologous. It is shown that all such knots are grid number 1 (also now often called simple knots) and tunnel number 1. The second theorem classifies such knots where the exterior of the knot is a once-punctured torus bundle. The monodromy of the bundle is given in each case.NEWLINENEWLINEKey to the proofs is studying Scharlemann cycles, which are particular cycles in graphs defined, in this case, from a knot, a once-punctured torus and a Heegaard torus. Scharlemann cycles of length 2 or 3 are of particular importance.NEWLINENEWLINENote that, at some points, proofs of results rely on material that occurs later in the paper. In particular, the proofs of the main theorems are given in Section 1.