12.3 Plumbing Fixture
259
(a)
(b)
(c)
(d)
Fig. 12.5 Different representations of the surface 1,2 obtained after gluing 1,1 and 0,3
through the plumbing fixture. (a) Direct gluing of circles. (b) Connection by a long tube. (c)
Insertion of the second surface into the first one. (d) Insertion of the first surface into the second
one
The first interpretation is the most direct one: the disks are simply removed and
the boundaries are glued together by a small tube of radius |w
(1)
1 | < |q| 1/2 . The
connection between both surfaces can be smoothed (and figures are often drawn
in this way—for example Fig. 12.6), but this is not necessary (the smoothing is
achieved by cutting disks of size |q| 1/2 − and gluing the boundaries to the disks
of radius |q| 1/2 ).
In the second interpretation, one rescales the local coordinates in order to bring
the radius of the disk to 1 instead of |q| 1/2 . In terms of these new coordinates, the
surfaces are connected by a tube of length s = − ln |q| after a Weyl transformation
(see [2, sec. 9.3] for a longer discussion).
259
(a)
(b)
(c)
(d)
Fig. 12.5 Different representations of the surface 1,2 obtained after gluing 1,1 and 0,3
through the plumbing fixture. (a) Direct gluing of circles. (b) Connection by a long tube. (c)
Insertion of the second surface into the first one. (d) Insertion of the first surface into the second
one
The first interpretation is the most direct one: the disks are simply removed and
the boundaries are glued together by a small tube of radius |w
(1)
1 | < |q| 1/2 . The
connection between both surfaces can be smoothed (and figures are often drawn
in this way—for example Fig. 12.6), but this is not necessary (the smoothing is
achieved by cutting disks of size |q| 1/2 − and gluing the boundaries to the disks
of radius |q| 1/2 ).
In the second interpretation, one rescales the local coordinates in order to bring
the radius of the disk to 1 instead of |q| 1/2 . In terms of these new coordinates, the
surfaces are connected by a tube of length s = − ln |q| after a Weyl transformation
(see [2, sec. 9.3] for a longer discussion).