Let K be a hyperbolic, fibered knot in
.
Then the exterior is regarded as a mapping torus
of a compact, once punctured surface with a monodromy
isotopic to a pseudo-Anosov automorphism.
Performing a longitudinal surgery on K,
we obtain a 3-manifold which is naturally regarded as
the mapping torus of the capped off surface
with the capped off monodromy;
the dual knot
(the core of the filled solid torus in the resulting 3-manifold)
is a section for the surface bundle.
Generically the resulting 3-manifold is still hyperbolic,
in other words,
the capped off monodromy is still isotopic to a pseudo-Anosov automorphism.
Gabai found a hyperbolic, fibered knot in
on which a longitudinal surgery produces a toroidal manifold,
and now it is known that
there are infinitely many such hyperbolic, fibered knots.
On the other hand,
there have been no known examples of hyperbolic, fibered knots
in
with longitudinal, Seifert fibered surgeries, and
Teragaito asks if there are no such examples.
We give an answer this question by constructing
an infinite family of hyperbolic, fibered knots in
each of which admits a longitudinal, Seifert fibered surgeries.
Besides our examples show
existence of boundary slopes which can be also Seifert fibered slopes.
We also give a condition assuring that
the given section in a Seifert fibered, surface bundle over the circle
is hyperbolic in terms of the "projection" in the fiber surface.