Suppose alpha and R are disjoint simple closed curves in the boundary of a genus two handlebody H such that H[R] (i.e. a 2-handle addition along R) embeds in S3 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 [J. Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003) 435-472.]. In this paper, we show that there exists a meridional curve m of k (or H[R]) in partial derivative 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 primitive/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 S3 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.
