Graphics Reference
In-Depth Information
permissible. Furthermore, because it is clearly easy to pass back and forth between
symbols and labeled polygons
Q
k
, we can use either of these two representations
interchangeably.
We have defined symbols for surfaces, but that is only half the story. We basically
have a “map” from the set of surfaces to the set of strings W. To really take advantage
of this correspondence we need a map that goes the other way, that is, we need a map
that associates a surface to a string in W. It is clear how to define this map on an intu-
itive level. Rather than cutting we shall paste. Basically, if w = a
1
a
2
...a
k
ŒW, then
we start with
Q
k
and construct a space
S
w
by pasting together any edges
e
i
(k) and
e
j
(k) of
Q
k
whenever a
i
= a
j
or a
j
-1
. It is easy to see that if w is an arbitrary string, then
the space
S
w
need not be a surface. Therefore, in addition to explaining the con-
struction of
S
w
more carefully, one would also like to know under what conditions
S
w
will be a surface.
Definition.
If a ŒS, then define n
w
(a) to be the number of times that the symbol a
or a
-1
appears in the string w. The
length of w
, l(w), is defined by
Â
()
=
()
1w
n
a
.
w
a
ŒÂ
For example, if w = A
1
A
2
A
-1
A
1
A
2
A
-1
, then n
w
(A
1
) = 3, n
w
(A
2
) = 2, n
w
(A
3
) = 1, and
n
w
(A
i
) = 0 for i > 3. Also, l(w) = 6.
Definition.
Define a subset W* of W by
{
()
=
}
W
*
=Œ
w
W
n
a
02
or
for all a
ŒÂ
.
w
It is easy to see that if w
S
is a symbol for a surface
S
, then w
S
ŒW*. In fact, the
next lemma shows among other things that
S
w
is a surface if and only if w ŒW*.
6.5.4. Lemma.
There is a construction that associates to each w ŒW* a well-defined
labeled complex (L
w
,m
w
) with the following properties:
(1) ÔL
w
Ô=
Q
l(w)
if l(w) > 2 and ÔL
w
Ô=
Q
6
if l(w) = 2.
(2) The space
S
w
= X
(L
w
,m
w
)
is a surface.
(3) If
S
is a surface and if u is any symbol for
S
, then
S
S
u
.
(4) Let a, b ŒS. If w = aa
-1
, then
S
w
S
2
. If w = aa, then
S
w
P
2
. If w = aba
-1
b
-1
,
S
1
¥
S
1
.
then
S
w
Proof.
See [AgoM76]. It is easy to justify part (4). Cutting a sphere along an arc
allows us to flatten the remainder into a disk with two edges that are appropriately
identified. For
P
2
, recall the discussion in Section 3.4 and Figure 3.9. For
S
1
¥
S
1
, see
Figure 6.17.
Definition.
If
S
is a surface, then any w ŒW* such that
S
w
S
will be called a
symbol
for
S
.
The fact that this new definition of a symbol for a surface is compatible with the
earlier one follows from Lemma 6.5.4(3).
