Graphics Reference
In-Depth Information
x
y
A
1
-1
A
2
A
1
x'
A
2
A
2
A
1
A
2
A
1
A
1
y'
-1
A
3
A
2
A
2
A
3
A
2
A
3
A
3
-1
A
3
A
3
(a)
(b)
(c)
Figure 6.20.
Alternate labeled complexes for sphere.
-
1
1
23
3
1
-
-
2
1
AA A AA A
1
for
S
2
. Figure 6.19(c) shows the corresponding labeled polygon.
Example 6.5.3 listed one symbol for a sphere, but it is clear that a different choice
of edge labels could have produced the symbol
-
1
-
1
-
1
AAAAAA
4
.
416
6
1
As we stated earlier, symbols for surfaces are not unique. On the other hand, although
the labeled complex (L
0
,m
0
) does not determine a unique symbol for the sphere, we
leave it to the reader to convince him/herself that all symbols derived from (L
0
,m
0
) will
have the form
-
1
1
23
3
1
-
-
2
1
aa a aa a
1
for a
i
ŒS. In general, one can show that every symbol associated to a labeled complex
(L
S
,m
S
) from Lemma 6.5.2 for a surface
S
has the same basic structure, that is, if a
1
a
2
...a
k
and b
1
b
2
...b
k
are two symbols associated to (L
S
,m
S
), then there is a permuta-
tion s of S such that b
i
=s(a
i
) and s(a
-1
) =s(a)
-1
for all a ŒS. This justifies our talking
about “the” symbol w
S
associated to (L
S
,m
S
) after all.
Continuing our sphere example, even though Figure 6.20(a) is a good pictorial
representation for one of its symbols, it is not the one that is usually adopted. By
drawing little arrows in the edges of
Q
6
as indicated in Figure 6.20(b), one can incor-
porate, without superscripts on the symbols, the same information that was contained
in Figure 6.20(a). The two possible ways of identifying edges (via linear maps) are
specified by the direction of the arrows. For example, the arrows tell us that the points
x
and
y
in Figure 6.20(c) are to be identified with the points
x
¢ and
y
¢, respectively.
Observe that the direction of the arrows is not uniquely specified by a symbol. Simul-
taneously reversing their direction on two edges that are to be identified changes
nothing. The only important property that is an invariant is whether these arrows
are both in the same or opposite direction. Most of the labeled polygons we shall
refer to from now on will have the arrows in their sides rather than superscripts on
the labels, but we should remember that either or both methods simultaneously is
