Hyperbolic tunnel-number-one knots with Seifert-fibered Dehn surgeries

Suppose $\alpha$ and $R$ are disjoint simple closed curves in the boundary of a genus two handlebody $H$ such that $H[R]$ embeds in $S^3$ as the exterior of a hyperbolic knot $k$(thus, $k$ is a tunnel-number-one knot), and $\alpha$ is Seifert in $H$(i.e., a 2-handle addition $H[\alpha ]$ is a Seifert-fibered space) and not the meridian of $H[R]$. Then for a slope $\gamma$ of $k$ represented by $\alpha$, $\gamma$-Dehn surgery $k(\gamma )$ is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was introduced in [D03]. In this paper, we show that there exists a meridional curve $M$ of $k$ (or $H[R]$) in $\mathrm{\partial }H$ such that $\alpha$ intersects $M$ transversely in exactly one point. It follows that such a construction of a Seifert-fibered Dehn surgery $k(\gamma )$ can arise from a primtive/Seifert position of $k$ with $\gamma$ its surface-slope. This result supports partially the two conjectures: (1) any Seifert-fibered surgery on a hyperbolic knot in $S^3$ is integral, and (2) any Seifert-fibered surgery on a hyperbolic tunnel-number-one knot arises from a primitive/Seifert position whose surface slope corresponds to the surgery slope.