Seifert fibered surgeries with distinct primitive/Seifert positions (Q409551)

Let \(V \cup _F W\) be a genus 2 Heegaard splitting of \(S^3\). In [Algebr. Geom. Topol. 3, 435--472 (2003; Zbl 1021.57002)], \textit{J. C. Dean} showed that if \(K \subset F\) is a curve such that (1) adding a 2-handle to \(V\) along \(K\) produces a solid torus, and (2) adding a 2-handle to \(W\) along \(K\) produces a Seifert fibered space over a disk, then Dehn surgery on \(K\), with surgery slope given by \(F \cap \partial N(K)\), yields a small Seifert fibered space. In [J. Knot Theory Ramifications 21, No. 1, 12 p. (2012; Zbl 1236.57014)], \textit{B. J. Guntel} produced the first examples of knots \(K\) and \(K'\) on \(F\) as above that were isotopic in \(S^3\), but not on \(F\).NEWLINENEWLINE In the present paper, the authors produce many more such examples. These are all twisted torus knots, or knots obtained by the Montesinos trick. The paper closes with an interesting question: Does there exist a universal bound for the number of distinct positions of a knot \(K\) on \(F\), as described above?