# Anti-Gravity Warp Drive

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:

# Anti-dimensional space and … the Big Bang, Big Bang Inflation and Where the Anti-matter Is!

Inflation.
Anti-dimensional space singularity and normative space dimensionality connect.

Matter in dimensional space /anti-matter in anti-dimensional space connect and erupt in the Big Bang.
Rapid expansion of space, inflation, due to the Anti-matter/Matter connection.
And where is the anti-matter? In anti-dimensional space!

# Anti-Dimensional Space and the Standard Model

I don’t know how anti-dimensional space would affect the standard model.

# When the Singularities of Anti-Dimensional Space and Normative Space Connect

Fluctuations of anti-dimension phase.
A cluster forms as phased space becomes ever more dense.
Phased fluctuations gravitate toward a central point.

Why does this happen?
When fluctuations in normative dimensional space phase, there is relative equal and opposite definition of fluctuation in anti-dimensional space.
Anti-dimensional space forms a super cluster.
Normative dimensional dimension space form a super cluster.
The superclusters connect.
The Big Bang ensues. Two Big Bangs.
A Big Bang occurs in normative dimensional space.
A Big Bang occurs in anti-dimensional space.
One is potential of the other;
matter is the potential of anti-matter,
anti-matter is the potential of matter.

This process is iterative.
If there is unevenness as the phased fluctuations and anti-phased fluctuations gravitate toward a central point, the anti-matter dimensions and matter dimensions connect.
A Big Bang occurs.
The initial conditions may result in a Universe that collapses.

This process continues until the normative dimensional universe and the anti-dimensional universe collapse in a uniformly smooth manner.
Entropy is low.
The normative dimension and anti-dimensional dimension connect and erupt.

# The Pardaox of the First Moment: How Can Hawkings Be So Wrong …

What happened prior to the BB was ongoing.
Still ongoing to day.
And, so, detectable.

How can we be sure that something happened before the BB?

Let’s try to understand what definition is.
Definition is equal and opposite relativity.
What defines dimensions of space. Equal and opposite relativity.
Plus dimension relative to negative dimensions.
Equilibrium. Balance. Definition.
Before equilibrium.
Non.
Ongoing.
No time.
No position.
No relativity.
No definition.
Dimensions in Non-equilibrium fluctuate.
interact.
Fluctuations phase.
Gravitate.

If we Untangle dimensions of space see before the BB what would we ‘see’?
Can we detect non-equilibrium Ds? No.
Could we detect fluctuations phase. Yes.
Would we see what happened before the BB? Yes.

# Can a particle traverse anti-dimensional space to emerge light years from the point of origin?

Space Travel through Anti-dimensional space:

Normative dimensional space and anti-dimensional space connect.
If a particle passes from normative dimensional space through to anti-dimensional space which again connects with normative dimensional space, does it traverse a distance?

Can we travel through anti-dimensional space to come out somewhere else?

# A Proof of Anti-Dimensions: What Could Make A String Theorist Happier?

Proof that there are anti-dimensions?
We can’t find anti-matter.
That’s where it is. In anti-dimensional space.

This should make string theorists happy.
String predicts additional dimensions. So, this is consistent with predictions by string theory.