Graphics Reference
In-Depth Information
Let K be a simplicial complex of dimension n and, as before, let n
q
= n
q
(K) be the
number of q-simplices in K. For each q, assume that we have chosen an orientation
for all the q-simplices in K and let
{
[][]
[
]
}
q
q
q
+
S
q
=
ss s
1
,
,...,
2
n
q
be the set of these oriented q-simplices of K. The set S
+
will be a basis for the free
abelian group C
q
(K). Define integers e
ij
by the equation
n
q
[
]
=
[]
Â
q
+
1
q
q
∂
s
e
s
q
+
1
j
ij
i
i
=
1
and note that
q
q
Ï
Ì
Ó
¸
˝
˛
1
0
if is a face of
otherwise
s
s
e
i
q
i
j
+
1
=
.
j+1
]. The
qth
incidence matrix
E
q
, 0 £ q < dim K, of K is defined to be the (n
q
¥ n
q+1
)-matrix
The integer e
ij
is called the
incidence number
of [s
i
] and [s
Definition.
[
]
[
] [
]
q
+
1
q
+
1
q
+
+
1
s
s
L
s
1
2
n
q
1
[]
[]
q
q
q
q
Ê
ˆ
e
e
L
e
s
s
11
12
1n
1
q+1
=
()
=
Á
Á
Á
Á
˜
˜
˜
˜
E
q
q
e
q
q
q
q
L
e
e
e
ij
2
21
22
2n
q+1
M
M
M OM
L
[
]
q
s
q
q
q
e
e
e
Ë
¯
n
q
n1
n2
nn
q
q
q
q+1
whose rows and columns are indexed by the elements of S
+
and S
q+1
, respectively.
Suppose that K =·
v
0
v
1
v
2
Ò and that we have chosen the S
+
7.2.5.1. Example.
as
follows:
+
=
{
}
S
S
S
vvv
vv
,, ,
,
012
0
+
=
[
{
] [
] [
]
}
vv
,
vv
,
01
12
0 2
1
+
=
[
{
]
}
vvv
.
012
2
The incidence matrices of K are then given by
[
]
[
]
[
]
[
]
vvv
vv
vv
vv
012
01
12
0 2
[
]
vv
vv
vv
+
+
-
1
1
1
v
v
v
+
1
0
-
1
Ê
ˆ
Ê
ˆ
01
0
1
0
E
=
and
E
=
Á
Á
˜
˜
Á
Á
˜
˜
[
]
-
1
-
+
1
0
12
1
[
]
Ë
¯
Ë
¯
.
0
1
+
1
02
2
