Genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings

We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into T×I (T is a torus with one boundary component). This condition provides a criterion whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered knot) which induces the Heegaard splitting. By using this, we determine the number and the positions with respect to the Heegaard splittings of GOF-knots in the 3-manifolds with reducible genus two Heegaard splittings. This gives an alternative and unified proof of the results of Morimoto [12] and Baker [2], [3].