http://www.math.cornell.edu/~mec/2008-2009/HoHonLeung/page5_knots.htm

Definition 5.2 Suppose K is an oriented knot (or link) and D is a (oriented) regular diagram for K. Then the Jones polynomial of K can be defined uniquely from the following two axioms. The polynomial itself is a Laurent polynomial in the square root of t, that is, it may have terms in which the square root of t has a negative exponent. This polynomial is a knot invariant for K.

http://www.math.cornell.edu/~mec/2008-2009/HoHonLeung/equation…19.jpg

2.3.3 Kasparov product [i

∗

] ⊗C(G/T)

[i!

]

Following the definition of Kasparov product, we can get the following:

[i

∗

] ⊗C(G/T)

[i!

] = [C(G/T) ⊗m L

2

(G/T, S), i, 1 ⊗ D]

where C(G/T) ⊗m L

2

(G/T, S), as an internal tensor product of two Hilbert modules,

is viewed as a G-Hilbert space. G acts on it by

g.(f ⊗m h) = (g.f) ⊗m (g.h)

where g ∈ G, f ∈ C(G/T) and h ∈ C

∞(G/T, S). We can extend this action to an

action on C(G/T) ⊗m L

2

(G/T, S) by continuity. i is the scalar multiplication on

C(G/T) ⊗m L

2

(G/T, S).

In general, the Kasparov product is hard to compute. But in our particular

case, Kasparov [K2] showed the following result:

Theorem 39 Let G be a compact group and M be a compact G-manifold. Let

[E] ∈ K0

G(M) be an element in the equivariant K-theory of M and let [

¯∂M] ∈

KKG(C(M), C) ∼= KG

0

(M) be the equivariant Dolbeault element. Then

[E] ⊗C(M)

[D] = G-index(( ¯∂M)E)

48

where (

¯∂M)E is the Dolbeault operator with coefficient in E.

Remark 40 If D is, say, an order-zero elliptic operator and E is a complex vector

bundle over a compact manifold M. In general it is permissible that D acts on

sections of bundles like the Dolbeault operator. But for the sake of notational

simplification we pretend that D acts on functions. We should think of D as a

bounded operator, by some basic functional calculus, on L

2

(M). Then we can

construct DE as an operator

DE : L

2

(M, E) −→ L

2

(M, E)

acting on sections of E. In general we define DE by using the local triviality of E

together with a partition of unity argument. Thus we choose a partition of unity

{f1, …, fk} for M such that each fi

is supported within an open set Ui over which

the bundle E is trivializable. Choosing trivializations and hence isomorphisms

L

2

(Ui

, E|Ui

) ∼= L

2

(Ui) ⊗ C

k where k is the dimension of the bundle, we define

operators (f

1/2

i Df 1/2

i

)E on L

2

(Ui

, E|Ui

) by pulling back the operators f

1/2

i Df 1/2

i ⊗1

on L

2

(Ui) ⊗ C

k via these isomorphisms. Finally we define DE to be the operator

DE =

X

k

i=1

(f

1/2

i Df 1/2

i

)E

on L

2

(M, E). The operator we obtain in this way depends on the choice of partition

of unity. However, whatever the choices DE is a Fredholm operator and its index

does not depend on the choices. In this way we obtain an index ind(DE) ∈ Z

for every [E] ∈ K0

(M). In the equivariant case where G is compact, DE is then

a G-equivariant Fredholm operator for [E] ∈ K0

G(M). The kernel and cokernel

are now (finite-dimensional) G-vector spaces and hence we get the G-index G −

index(DE) ∈ R(G).

49

Topologically, the element [i

∗

] ∈ KKG(C, C(G/T)) ∼= K0

G(C(G/T)) corresponds

to the trivial G-bundle E0 over G/T. The homogeneous pseudo-differential

operator DE0 has G-index 1G ∈ R(G) by a result of Bott, see [Bo]. By Theorem

39, we have the following result:

Theorem 41 [i

∗

] ⊗C(G/T)

[i!

] = 1 ∈ KKG(C, C)

**https://ecommons.cornell.edu/bitstream/handle/1813/29318/hl363thesisPDF.pdf?sequence=1**

Example 10 Using the same notations as in the example in 1.4.2, G

(12)

(23) = (1 −

y3

x1

) ∈ K∗

T

(Fl(C

3

)). There are six fixed points for each element in S3,

G

(12)

(23)|(23) 6= 0, G(12)

(23)|(123) 6= 0, G(12)

(23)|(13) = 0

G

(12)

(23)|(132) = 0, G(12)

(23)|(12) 6= 0, G(12)

(23)|id 6= 0

So the support of a permuted double Grothendieck polynomial contains

id,(12),(23),(123). On the other hand,

(X

(12)

(23) )

T = {v ∈ S3 | (12)v ≤ (12)(23) = (123)}

= {v ∈ S3 | (12)v ≤ id,(12),(23) or (123)}

= {v ∈ S3 | v ≤ (12), id,(123) or (23)}

which is the same as Supp(G

(12)

(23)).

Now we show a fundamental relation between the permuted double

Grothendieck polynomials and the permuted Bruhat Orderings: