Drorbn - User contributions [en]

Notes for AKT-091008-1/0:51:07

2011-11-08T10:40:42Z

Iva:

Given a planar diagram, number of 3-colorings of the edges = number of 4-colorings of the regions in the plane divided by the diagram.

Bijection using the Klein 4 group as colors and add when crossing edges.

Notes for AKT-091008-1/0:35:51

2011-11-08T02:11:26Z

Iva:

Translate of the statement:

Every planar diagram has a 4-coloring.

Notes for AKT-091008-1/0:31:58

2011-11-08T02:11:13Z

Iva:

''Comments'':

# The statement is reasonable and doesn't look hard.
# <math>|W_{sl_N}^{top}(D)|</math> is proportional to the number of planar embeddings of <math>D</math> with minor conditions.
# If <math>D</math> is planar, <math>|W_{sl_2}(D)|</math> is proportional to the number of 4-colorings of the plane divided by <math>D</math>.
#The statement above is equivalent to the four colour theorem.

Notes for AKT-091008-1/0:31:58

2011-11-03T13:32:03Z

Iva:

Notes for AKT-091008-1/0:23:59

2011-11-03T13:30:12Z

Iva:

''Statement'':

:" <math>W_{sl_2}(D)=0 \ \Rightarrow \ W_{sl_N}^{top}(D)=0</math> "

where <math> W_{sl_N}^{top}(D) = \mbox{Coeff}_{N^{\deg{D}+2}}(W_{sl_N}(D))</math>.

Notes for AKT-091008-1/0:21:48

2011-11-03T04:45:09Z

Iva:

''Aside'': <math>W_{\mathcal{G}}</math> (without the representation) makes sense on <math>\mathcal{A}(\phi)</math> (trivalent diagrams without skeleton) with the option to mod out by the AS and IHX relations.

Notes for AKT-091008-1/0:19:04

2011-11-03T04:44:16Z

Iva:

''Comment'': <math>W_{gl_N, R^N}</math> is the HOMFLY weight system up to the framing independence relation. i.e.

:<math>W_{gl_N, R^N} \circ p (D) = W_H(D)</math>

Notes for AKT-091008-1/0:14:19

2011-11-03T04:27:34Z

Iva:

Sample diagram 4: the <math>D_{ai}</math> diagram

(The value being <math>n</math> raised to the power equal to the number of connected components after we replace all intersection points according to the recipe.)

Notes for AKT-091006/0:11:24

2011-10-30T16:20:45Z

Iva:

Choose a basis <math>(X_a)^{\dim{\mathcal{G}}}_{a=1}</math> for <math>\mathcal{G}</math>
and a basis <math>(e_{\alpha})^{\dim(R)}_{\alpha=1}</math> for <math>R</math>.

''Notation'':
:<math>[X_a, X_b] = f_{a,b}^c X_c</math>, where <math>f_{ab}^c \in \mathbb{Q}</math> are the structure constants

:<math>\left\langle X_a, X_b\right\rangle = t_{ab}</math>
::Symmetric: <math>t_{ab}=t_{ba}</math>
::Non-degenerate: <math>(t_{ab})</math> has an inverse, <math>(t^{ab})</math>, with <math>t_{ab} \cdot t^{bc} = \delta_{ac}</math>

Notes for AKT-091006/0:20:44

2011-10-30T16:18:09Z

Iva:

Expression representation, Jacobi identity and "<math>r \cdot f = r_1 r_2 - r_2 r_1</math>"

Notes for AKT-091006/0:11:24

2011-10-30T16:02:04Z

Iva:

Choose a basis <math>(X_a)^{\dim{\mathcal{G}}}_{a=1}</math> for <math>\mathcal{G}</math>
and a basis <math>(e_a)^{\dim(R)}_{a=1}</math> for <math>R</math>.

''Notation'':
:<math>[X_a, X_b] = f_{a,b}^c X_c</math>, where <math>f_{ab}^c \in \mathbb{Q}</math> are the structure constants

:<math>\left\langle X_a, X_b\right\rangle = t_{ab}</math>
::Symmetric: <math>t_{ab}=t_{ba}</math>
::Non-degenerate: <math>(t_{ab})</math> has an inverse, <math>(t^{ab})</math>, with <math>t_{ab} \cdot t^{bc} = \delta_{ac}</math>

Notes for AKT-091006/0:08:29

2011-10-30T15:45:07Z

Iva:

Overview: we are now going to summarize the Lie-algebraic information in terms of tensors.

Warning: there will be many indices.

(Note: This can also be done abstractly, i.e. without choosing a basis, but then we can`t calculate anything.)

Notes for AKT-091006/0:03:35

2011-10-30T15:33:03Z

Iva:

Example: Let the Lie algebra be <math>gl_n(k)</math> (with <math>[ \cdot, \cdot]</math> being the commutator and <math>\left\langle A, B \right\rangle= tr(AB)</math>)

Let the representation <math>R</math>: <math>\rho: \mathcal{G} \rightarrow End(R), \rho([A, B]) = \rho(A) \circ \rho(B) - \rho(B) \circ \rho(A)</math>

Notes for AKT-091006/0:00:02

2011-10-28T02:33:33Z

Iva:

Where we are in the diagram: usual knots, low algebra (Lie-algebra)

''Goal'': Given a finite-dimensional, metrized Lie algebra <math>\mathcal{G}</math> and a finite-dimensional representation <math>R</math>, we will construct a functional <math>W_{\mathcal{G},R}: A(O) \rightarrow k</math> (the base field).

Notes for AKT-091001-2/0:15:43

2011-10-27T12:22:04Z

Iva:

2) Given <math>\mathcal{G}</math> (metrized finite dim Lie algebra) <math>\exists\ {\mathcal T}_\mathcal{G}: \mathcal{A}(|) \rightarrow U(\mathcal{G})</math> where <math>U(\mathcal{G})</math> is the universal enveloping algebra of <math>\mathcal{G}</math>.

Notes for AKT-091001-2/0:13:25

2011-10-27T12:21:34Z

Iva:

'''Theorem''':

1) Given a metrized, finite-dimensional Lie algebra <math>\mathcal{G}</math> and a finite-dimensional representation <math>R</math> of <math>\mathcal{G}</math>, <math>\exists</math> (an interesting) linear functional <math>W_{\mathcal{G},R}: \mathcal{A} \rightarrow \mathbb{Q}</math> (the ground field on which <math>\mathcal{G}</math> is defined).

Notes for AKT-091001-2/0:09:23

2011-10-19T13:17:11Z

Iva:

'''Corollary''': Diagrams with only one connected component (after removing the outer circle) are primitive.
In fact one can show those diagrams span the space of primitives.

Notes for AKT-091001-2/0:05:19

2011-10-19T13:06:20Z

Iva:

Verify that the co-product satisfies the STU relation.

Notes for AKT-091001-2/0:01:11

2011-10-19T13:05:46Z

Iva:

Before that, some combinatorics: <math>\mathcal{A}^t</math> is a bialgebra with

# Product: concatenation.
# Coproduct: sum over all ways of distributing the connected components, when the skeleton is ignored, between the two sides of the tensor product.

Notes for AKT-091001-2/0:00:25

2011-10-19T12:50:30Z

Iva:

'''Aim''': Make the relationship between <math>\mathcal{A}</math> and Lie algebras formal and move into low algebra.

Notes for AKT-091001-1/0:03:45

2011-10-19T05:13:53Z

Iva:

'''Goal''': Describe the relationship between <math>\mathcal{A}</math> and Lie algebras.

Notes for AKT-090929/0:46:15

2011-10-19T04:40:42Z

Iva:

Comment by Peter about the map <math>P</math>.

Notes for AKT-090929/0:46:15

2011-10-19T04:40:19Z

Iva:

Comment by Peter.

Notes for AKT-090929/0:42:53

2011-10-19T04:09:55Z

Iva:

We can deduce that the projection <math>P:\mathcal{A} \rightarrow \mathcal{A}</math> which takes <math>\theta</math> to zero is given by:

:<math>P=\sum^{\infty}_{n=0}\frac{(-\hat{\theta})^n}{n!}(\hat{W}^*_1)^n</math>

Notes for AKT-090929/0:37:23

2011-10-19T04:06:56Z

Iva:

Relation to quantum mechanics (Von Neumann's theorem): whenever the commutator of two operations equals the identity, one should be thought of as the derivative and the other as multiplication. Here the two operators are <math>\hat{\theta}</math> - multiplication by <math>\theta</math>, and <math>\hat{W}^*_1</math> - the adjoint of the dual of <math>\hat{\theta}</math>.

Notes for AKT-090929/0:09:24

2011-10-19T03:48:41Z

Iva:

'''Theorem (Milnor-Moore)''': A graded, connected, co-commutative bialgebra is the universal enveloping algebra of its space of primitives:
: <math>\mathcal{A}=U(\mathcal{P}(\mathcal{A}))</math>

(A proof is given [http://math.uchicago.edu/~mitya/bloch-hopf/hopf3.pdf here].)

Notes for AKT-091001-1/0:04:11

2011-09-25T18:17:03Z

Iva:

Introducing a new pictorial presentation <math>\mathcal{A}^t</math> of <math>\mathcal{A}</math> in order to introduce the relationship between <math>\mathcal{A}</math> and Lie algebras (t stands for trivalent vertex).

Notes for AKT-090929/0:09:24

2011-09-25T15:54:18Z

Iva:

'''Theorem (Milnor-Moore)''': A graded, connected, co-commutative bialgebra is the universal enveloping algebra of its space of primitives:
: <math>\mathcal{A}=U(\mathcal{P}(\mathcal{A}))</math>

Notes for AKT-090929/0:10:43

2011-09-25T15:51:10Z

Iva:

'''Definition.''' Primitive:
<math>\mathcal{P}(\mathcal{A})=\{a \in \mathcal{A}: \square a= a \otimes 1 + 1 \otimes a\}</math>, where <math>\square</math> denotes the coproduct.

Notes for AKT-090929/0:09:24

2011-09-25T15:47:41Z

Iva:

'''Theorem (Milnor-Moore)''': A graded, connected, co-commutative bialgebra is the universal enveloping algebra of its space of primitives.

Notes for AKT-090924-2/0:50:15

2011-09-17T21:07:57Z

Iva:

'''Claim''': <math>\mathcal{A}</math> is a bi-algebra.

Notes for AKT-090924-2/0:46:01

2011-09-17T21:06:53Z

Iva:

''Definition'': A bialgebra <math>B</math> is a <math> B</math> with an algebra and a co-algebra structure such that the co-algebra operations (comultiplication and counit) are morphisms of algebras.

Notes for AKT-090924-2/0:39:41

2011-09-17T20:45:19Z

Iva:

Hints:

1. Leibniz rule for derivative of products:

:<math>\partial_x(f.g)=(\partial_x f) g +f (\partial_x g)</math>

2. Iterated Leibniz rule (for expressions like <math>(\partial_x \partial_y \partial_z) (f.g)</math>)

3. Leibniz rule in a discrete (combinatorial) setting:

<math>(f.g)(\doublepoint)=f(\doublepoint)g(\overcrossing) + f(\undercrossing)g(\doublepoint)</math>

Notes for AKT-090924-2/0:31:10

2011-09-17T20:13:37Z

Iva:

'''Claim''': The co-multiplication is well-defined (check that it is constant for diagrams differing by a <math>4T</math> relation).

Notes for AKT-090924-2/0:24:38

2011-09-17T16:56:13Z

Iva:

'''Claim''': Our space <math>\mathcal{A}</math> is a co-commutative co-algebra (under the co-multiplication and co-identity as defined, both of which are analogous to the polynomial example).

Notes for AKT-090924-2/0:00:18

2011-09-15T13:14:35Z

Iva:

Deduce (from the two pictures given last time) that <math>\mathcal{A}</math> is a commutative algebra.

Notes for AKT-090924-1/0:35:01

2011-09-15T13:11:42Z

Iva:

Analog for <math>\mathcal{A}</math>: <math>\mathcal{A}(\bigcirc)</math> is isomorphic to <math>\mathcal{A}(|)</math>.

(Checking the map of 'breaking the circle' is independent of the choice of breaking point is interesting.)

Notes for AKT-090924-1/0:30:11

2011-09-15T13:08:48Z

Iva:

'''Claim 1''': There is a bijection between round knots (i.e. knots on a circle) and long knots (i.e. knots on a long line):
: <math>\mathcal{K}(\bigcirc)=\mathcal{K}(|)</math>

'''Claim 2''': <math>\mathcal{K}(|)</math> is an abelian monoid.

''Remark'': I think it's worth checking that the map from circle to line is independent of the choice of point to 'open up' and the path we 'pull out' the two ends after cutting. However it is indeed independent.

Notes for AKT-090924-1/0:21:04

2011-09-15T13:02:59Z

Iva:

Type <math>n</math> invariants for <math>n=0,1,2</math>:
# Type <math>0</math>: Constant
# Type <math>1</math>: Writhe<math>=W</math>
# Type <math>2</math>: <math>W^2</math>, <math>J_2</math> (The second coefficient in the expansion of <math>J(e^x)</math> where <math>J</math> is the Jones polynomial.)

Notes for AKT-090924-1/0:25:49

2011-09-15T12:56:37Z

Iva:

<math>\mathcal{A} = \hat{\bigoplus} \mathcal{A}_n</math> is (in some sense) the double dual of the space of knots

'''Theorem''': <math>\mathcal{A}</math> is a commutative co-commutative bi-algebra.
(For now, we prove <math>\mathcal{A}</math> is a commutative algebra.)

Notes for AKT-090924-1/0:07:21

2011-09-15T12:54:57Z

Iva:

The line <math>\mathcal{A}_n^\star = \mathcal{A}_n /{4T}</math> is a misprint and should in fact be <math>\mathcal{A}_n = \mathcal{D}_n /{4T}</math>.

Notes for AKT-090924-1/0:05:49

2011-09-15T12:54:12Z

Iva:

''Correction'': let <math>\mathcal{D}_n</math> be the space freely generated by all chord diagrams with <math>n</math> chords (instead of being just the set of all such diagrams)

Restating of the fundamental theorem: <math>W_n = {\mathcal{A}^\star_n}</math> where <math>\mathcal{A}_n = \mathcal{D}_n /{4T}</math>

Notes for AKT-090924-1/0:24:07

2011-09-15T12:53:07Z

Iva:

''Remark'': We do not have a generating function for the dimension of <math>\mathcal{A}_n</math>.

Notes for AKT-090924-1/0:07:33

2011-09-15T12:52:15Z

Iva:

We can also express <math>\mathcal{A}_n=\mathcal{D}^0_n/b(\mathcal{D}^1_n)</math> where <math>\mathcal{D}^0_n</math> is the v.s. of chord diagrams with <math>n</math> chords, <math>\mathcal{D}^1_n</math> is the v.s. of chord diagrams with one <math>T</math> shape and <math>(n-2)</math> chords and <math>b</math> is the map that takes the <math>T</math> shape to the signed sum of its four resolutions as in the 4T relation.

Notes for AKT-090924-1/0:05:49

2011-09-15T12:46:08Z

Iva:

Correction: let <math>D_n</math> be the space freely generated by all chord diagrams with <math>n</math> chords (instead of being just the set of all such diagrams)

Restating of the fundamental theorem: <math>W_n = {A^\star_n}</math> where <math>A_n = D_n /{4T}</math>

Notes for AKT-090922/0:49:35

2011-09-15T10:18:34Z

Iva:

''Remark'': There is a Reidemeister theory for framed knots as well, where the R2 and R3 moves are allowed but not the R1 move. The writhe is an invariant of such knots.

Notes for AKT-090922/0:49:35

2011-09-15T10:11:09Z

Iva:

''Remark'': There is a Reidemeister theory for framed knots as well were the R2 and R3 moves are allowed but not the R1 move. The writhe is an invariant of such knots.

Notes for AKT-090922/0:51:57

2011-09-15T10:10:31Z

Iva:

Analogously, there are finite type invariants for framed knots and we have:

'''Theorem''': Any <math>W</math> satisfying the 4T relation comes from an invariant <math>V</math> of framed knots.

Notes for AKT-090922/0:49:35

2011-09-15T10:06:12Z

Iva:

''Remark'': There is a Reidemeister theory for framed knots as well were the R2 and R3 moves are allowed but not the R1 move.

Notes for AKT-090922/0:45:17

2011-09-15T10:03:40Z

Iva:

We have the short exact sequence:

:<math>0 \rightarrow \mathbb{Z} \rightarrow \{</math>framed knots<math>\} \rightarrow \{</math>knots<math>\} \rightarrow 0</math>

with a splitting on the left given by the writhe.