Neurology Mathematics Ideas125.pdf - Example C.6 Consider...

Example C.6. Consider the fillable disk
A
2
M
0
1
1
D =
(C.25)
Using Equation (3.45) and Lemma 3.17 we obtain
℧
D
=
integraldisplay
m ∈M
T
[0]
( m ⊠ D
A
) ⊠ m =
integraldisplay
m ∈M
integraldisplay
a ∈A
a ∨ .m ⊠ D. [3] a ⊠ m
∼
=
integraldisplay
m ∈M
integraldisplay
a ∈A
a
∨
.m ⊠ a ⊠ m
∼
=
integraldisplay
m ∈M
integraldisplay
a ∈A
m ⊠ a ⊠ a.m.
(C.26)
We have actually already encountered the defect one-manifold that constitutes the outer bound-
ary of the disk D : it is the defect circle I
ւ
κ
( M ) in (5.14) with framing index κ = 0. Similarly
we obtain the following list of defect one-manifolds and silent objects for all other transparent
disks with outer boundaries given by one of the circles (5.14):
I
ր
κ
( M ) : ℧ =
integraldisplay
m ∈M
integraldisplay
a ∈A
m ⊠ m.a
[ κ − 1]
⊠ a =
integraldisplay
m ∈M
integraldisplay
a ∈A
m ⊠ m.a ⊠
[ κ − 1]
a,
I
ւ
− κ
( M ) : ℧ =
integraldisplay
m ∈M
integraldisplay
a ∈A
m ⊠ a ⊠ a
[ κ ]
.m,
I
տ
κ
( M ) : ℧ =
integraldisplay
m ∈M
integraldisplay
a ∈A
m ⊠ a ⊠
[ κ − 1]
a.m,
I
ց
− κ
( M ) :