Graphics Reference
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not intersect in a simplex
in
A. On the other hand, ÔAÔ is the underlying space of the
simplicial complex K in Figure 6.11(b). A useful fact is the following:
6.3.1. Proposition.
Let K be a simplicial complex. Every point of the underlying
space ÔKÔ belongs to the interior of a unique simplex of K.
Proof.
Exercise 6.3.1.
Definition.
Let K be a simplicial complex. A
subcomplex
of K is a simplicial complex
L with L Õ K.
For example, in Figure 6.10 the set M = {
v
0
,
v
2
,
v
0
v
2
} is a subcomplex of L. M is not
a subcomplex of K even though ÔMÔÕÔKÔ.
Definition.
The
boundary
of a simplicial complex K, denoted by ∂K, is defined by
{
k
∂=
K
tt
is a face of a simplex
s
Œ
K that belongs
(
)
}
to a
unique
k
+
1
-
simplex of K
.
It is easy to see that ∂K is a subcomplex of K. For example, if K is the simplicial
complex in Figure 6.9(b), then
∂=
{
}
K
vvvvvvvvvv
,, ,,
,
,
.
0 1 2 3 12 23 13
This example also shows that the underlying space of ∂K, Ô∂ KÔ, is not necessarily the
boundary of ÔKÔ because, thinking of ÔKÔ as a subset of
R
2
,
bdry K
=∂
K
»
v
01
.
We always have Ô∂ KÔÕbdry ÔKÔ and the two sets are the same if ÔKÔ is an n-
dimensional manifold in
R
n
.
Definition.
Let K be a simplicial complex and let
v
and
w
be vertices of K. An
edge
path
in K from
v
to
w
is a sequence of vertices
v
=
v
0
,
v
1
,...,
v
n
=
w
of K with the
property that
v
i
v
i+1
is a 1-simplex in K for 0 £ i < n. An edge path from
v
to
v
is called
an
edge loop
at
v
.
Definition.
A simplicial complex K is
connected
if, given any two vertices
v
and
w
in K, there is an edge path from
v
to
w
.
The simplicial complex in Figure 6.9(a) is
not
connected, whereas those in Figures
6.9(b) and 6.9(c)
are
. It is easy to show that a simplicial complex K is connected if
and only if ÔKÔ is path-connected.
Definition.
Let
X
be a subspace of
R
n
. A
triangulation
of
X
is a pair (K,j), where K
is a simplicial complex and j : ÔKÔÆ
X
is a homeomorphism. The complex K is said
to
triangulate
X
. A (finite)
polyhedron
is any space that admits a triangulation.
