Graphics Reference
In-Depth Information
e
1
(k) of
Q
k
with an arbitrary element from S. Continue around to the other edges of
Q
k
in a counterclockwise fashion and associate a different label from S to each of them (if
“a” has been used, “a
-1
” does not count as different)
unless
the new edge, say
e
i
=
e
i
(k),
is identified with a previously labeled edge, say
e
j
=
e
j
(k). In that case, the labels for
e
i
and
e
j
should reflect this identification while at the same time distinguishing between
the two possible ways that the edges could be identified. Assume that
e
j
has been
labeled “a.” If the edges are identified in an orientation-preserving way, then use the
label “a” for
e
i
, otherwise use “a
-1
.”
A more rigorous recursive definition of the string w
S
is the following. Define a
1
to
be an arbitrary element of S. Let 2 £ i £ k and assume that the elements a
1
, a
2
,...,
and a
i-1
have already been defined. The definition of a
i
divides into two cases:
Case 1.
y
S
(
e
i
(k)) πy
S
(
e
j
(k)) for all j, 1 £ j < i : In this case, let a
i
be an arbitrary
element of
S-
{
}
1
1
-
-
2
1
-
1
aa a a
,
,
,
,
K
,
a
i
1
2
-
Case 2.
y
S
(
e
i
(k)) =y
S
(
e
j
(k)) for some j, 1 £ j < i : In this case, let
(
()
)
=
(
()
)
aaif
=
y
w
k
y
w
k
i
j
S
i
-
1
S
j
-
1
-
1
=
a
otherwise
.
j
Definition.
The string w
S
ŒWis called a
symbol associated to (L
S
,m
S
)
, or simply a
symbol
for the surface
S
.
Note that the symbol for a surface is not unique since there is no unique choice
of a
i
in Case 1 above.
To find a symbol for the sphere
S
2
.
6.5.3. Example.
Suppose that we have triangulated
S
2
with the complex
Solution.
K =∂
vvvv
0123
,
where
v
0
v
1
v
2
v
3
is some 3-simplex. The first task is to find a labeled complex for some
regular k-gon of the type guaranteed by Lemma 6.5.2. Although this would not be
hard to do directly in our special case, in general it would be easiest to use two steps:
One would first flatten the surface out into the plane by cutting and then move the
result to a regular k-gon. We shall follow this general approach, which is actually how
Lemma 6.5.2 would be proved. See Figure 6.19(a). The first step would produce the
simplicial complex L with the simplicial map
a :L
Æ
K
defined by the condition that
a ss s
¢
:
¢ Æ
i
i
i
