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.