S
2
,
→
R
3
has no “knotting” behavior, because there are two few dimensions to move
around in, just like
S
1
,
→
R
2
.
But
S
2
,
→
R
4
works, and this is a special case of
higher
dimensional knot theory
:
S
n
,
→
R
n
+2
.
Think of
R
4
as 3dimensional space moving
through time, and think of
S
2
as a collection of circles. Now consider embeddings
S
2
⊂
R
4
as a “movie” of embeddings of
S
1
⊂
R
3
where unknots turn into knots and back to unknots.
20.4
Domain
X
More generally, there is knotting of objects
X ,
→
R
n
for appropriate
n
based on the dimension
of
X
. For example, tori can be knotted in
R
3
, since tori are nothing more than the boundaries
of thickened knots!
Exercise:
Find two 2holed tori that are not isotopic to each other in
R
3
.
32
21
4/12: Jones’ original paper
Backtrack to braid theory and the Jones polynomial. Let’s connect them, and in particular,
see how Jones originally defined his polynomial (before Kauffman did)!
Definition 18.
The
TemperleyLieb algebra
A
n
for
n >
1 is the free additive algebra on the
(multiplicative) generators
e
1
, . . . , e
n

1
viewed as a
C
[
τ, τ

1
]module.
Here
τ
is a variable
that commutes with all generators, and the generators satisfy the relations
1)
e
2
i
=
e
i
2)
e
i
e
i
±
1
e
i
=
τe
i
3)
e
i
e
j
=
e
j
e
i
when

i

j

>
2
Jones noticed that
A
n
looks like the Braid group
B
n
. He constructed a particular map
ρ
:
B
n
→
A
n
. He also constructed a particular “trace” map
tr
:
A
n
→
C
[
τ, τ

1
], which means
tr
(
ab
) =
tr
(
ba
). So we get a trace map
tr
◦
ρ
:
B
n
→
C
[
τ, τ

1
]
and this is the
Jones polynomial
after a particular normalization!
Let’s analyze this more closely. By Alexander’s theorem, any oriented link is given by the
closure ˆ
σ
of some braid
σ
. The nonuniqueness is handled by Markov’s theorem, so we have
to make sure that
tr
◦
ρ
is independent of the “Markov moves”, in order to get an invariant
of the link (closure of the braid). Let’s look at the Markov moves:
1) Vacuously, two braids are identified up to isotopyequivalence in the braid group.
2) Conjugate a braid
σ
∈
B
n
by another braid
b
∈
B
n
.
Exercise:
\
bσb

1
= ˆ
σ
3) An actual
Markov move
: Given
σ
∈
B
n
, view it inside
B
n
+1
by attaching an isolated
strand at the end, take the generator
σ
n
∈
B
n
+1
, and form
σσ
±
1
n
.
Exercise:
[
σσ
±
1
n
= ˆ
σ
4) The “inverse”
Markov move
, undoing (3), where a braid of the form
σσ
±
1
n
becomes
σ
.
The definition of “trace” already handles conjugation of braids.
To handle the actual
Markov moves, Jones’ trace map
tr
for
A
n
is chosen to satisfy the additional property
tr
(
we
i
) =
τ
·
tr
(
w
) whenever
w
∈
A
i

1
. To get our desired link invariant, we need to normalize
the trace map to account for these
τ
’s that appear under Jones’ additional property when a
Markov move occurs.
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 Spring '08
 Staff
 Math