アブストラクト

We say that a pair (K, m) of a knot K in the 3-sphere and
an integer m is a Seifert surgery if the result (K; m) of
the m-surgery on K is a Seifert fiber space;
here we allow a degenerate Seifert fiber space,
meaning (K; m) has a fiber of index zero.
All the known Seifert surgeries (K, m) have a knot c
disjoint from K such that c is unknotted in the 3-sphere and becomes
a Seifert fiber in the resulting Seifert fiber space (K; m).
We call such a knot as c a ``seiferter'' for (K, m).
As we have previously shown,
if an r-surgery on K yielding a Seifert fiber space for some rational number
r has a seiferter, then r is integral,
except when K is a torus knot or a cable of a torus knot.

Let us look each Seifert surgery as a ``vertex'',
and connect two vertices by an ``edge'' if they are related by
a single twisting along a seiferter.
This leads us to build a ``network'' of Seifert surgeries in which
each Seifert surgery is a vertex.

As we will observe,
many Seifert surgeries are connected to those on torus knots and
we expect that the network gives us a global picture of Seifert surgeries.
In this context, if we have a path in the network from a Seifert surgery (K, m)
to a known Seifert surgery (K_0, m_0), say K_0 being a torus knot,
then we can understand inductively (along the path)
how to obtain the given Seifert surgery.

We will establish some fundamental properties of the network.
In particular,
we show:
(1) Most seiferters for Seifert surgeries on hyperbolic knots
which become exceptional fibers
are shortest geodesics in the knot complements.
(2) The Berge's lens surgeries are ``close'' to
Seifert surgeries on torus knots.

From the networking viewpoint we show also:
(3) The network contains an infinite family of
Seifert fibered surgeries on hyperbolic knots which cannot be embedded
in genus 2 Heegaard surface of the 3-sphere.