Anti-Gravity Warp Drive

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.…19.jpg

2.3.3 Kasparov product [i

] ⊗C(G/T)
Following the definition of Kasparov product, we can get the following:

] ⊗C(G/T)
] = [C(G/T) ⊗m L
(G/T, S), i, 1 ⊗ D]
where C(G/T) ⊗m L
(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
(G/T, S) by continuity. i is the scalar multiplication on
C(G/T) ⊗m L
(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
(M) be the equivariant Dolbeault element. Then
[E] ⊗C(M)
[D] = G-index(( ¯∂M)E)
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
(M). Then we can
construct DE as an operator
DE : L
(M, E) −→ L
(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
, E|Ui
) ∼= L
(Ui) ⊗ C
k where k is the dimension of the bundle, we define
operators (f
i Df 1/2
)E on L
, E|Ui
) by pulling back the operators f
i Df 1/2
i ⊗1
on L
(Ui) ⊗ C
k via these isomorphisms. Finally we define DE to be the operator
DE =
i Df 1/2
on L
(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).
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)
] = 1 ∈ KKG(C, C)
Example 10 Using the same notations as in the example in 1.4.2, G
(23) = (1 −
) ∈ K∗
)). There are six fixed points for each element in S3,
(23)|(23) 6= 0, G(12)
(23)|(123) 6= 0, G(12)
(23)|(13) = 0
(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,
(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
Now we show a fundamental relation between the permuted double
Grothendieck polynomials and the permuted Bruhat Orderings:

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