Knots with the lens space surgery
In this paper a family of knots \(K_{m,n}\) is defined, where \(m\), \(n\) are positive integers. It is shown that each knot in this family is a hyperbolic strongly invertible \((1,1)\)-knot. An explicit presentation of the fundamental group of each of these knots is given, and then an explicit presentation is given of the fundamental group of the manifolds obtained by Dehn surgery on these knots. In the case that \(n=m+3\), it is shown that \(K_{m,m+3}\) has some exceptional Dehn surgeries, namely, it is shown that surgery with slopes \(9m+11\) and \(9m+7\) produces manifolds containing an incompressible torus but not containing a Klein bottle, surgery with slopes \(9m+6\) and \(9m+10\) produces Seifert fibered manifolds, while surgery with slope \(9m+8\) produces a lens space. This is done by a careful manipulation of a presentation of the corresponding fundamental groups. So, these knots have two toroidal Dehn surgeries at distance 4. However, this result gives a different description of a family of knots that was already known. \textit{C. McA. Gordon} and \textit{Y.-Q. Wu} have determined all hyperbolic knots with a pair of toroidal surgeries at distance 4 [Mem. Am. Math. Soc. 909, 140 p. (2008; Zbl 1166.57014)], and it follows that the knot \(K_{m,m+3}\) corresponds to the knot \(L_2(-m-1)\) as defined by Gordon and Wu.
scientific article; zbMATH DE number 6147205
scientific article