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AKT-140106 Video

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Notes on AKT-140106:    [edit, refresh]

Course introduction, knots and Reidemeister moves, knot colourings.

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# Week of... Notes and Links
1 Jan 6 About This Class (PDF).
dbnvp Monday: Course introduction, knots and Reidemeister moves, knot colourings.
Tricolourability without Diagrams
dbnvp Wednesday: The Gauss linking number combinatorially and as an integral.
dbnvp Friday: The Schroedinger equation and path integrals.
Friday Introduction (the quantum pendulum)
2 Jan 13 Homework Assignment 1.
dbnvp Monday: The Kauffman bracket and the Jones polynomial.
dbnvp Wednesday: Self-linking using swaddling.
dbnvp Friday: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
3 Jan 20 Homework Assignment 2.
dbnvp Monday: The definition of finite-type and some examples.
dbnvp Wednesday: The self-linking number and framings.
dbnvp Friday: Integrating a polynomial times a Gaussian.
Class Photo.
4 Jan 27 Homework Assignment 3.
dbnvp Monday: Chord diagrams and weight systems.
dbnvp Wednesday: Swaddling maps and framings, general configuration space integrals.
dbnvp Friday: Some analysis of d^{-1}.
5 Feb 3 Homework Assignment 4.
dbnvp Monday: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
dbnvp Wednesday: The Fulton-MacPherson Compactification, Part I.
dbnvp Friday: More on pushforwards, d^{-1}, and d^\ast.
6 Feb 10 Homework Assignment 5.
dbnvp Monday: The bracket-rise theorem and the invariance principle.
dbnvp Wednesday: The Fulton-MacPherson Compactification, Part II.
dbnvp Friday: Gauge fixing, the beginning of Feynman diagrams.
R Feb 17 Reading Week.
7 Feb 24 dbnvp Monday: A review of Lie algebras.
dbnvp Wednesday: Graph cohomology and \Omega_{dR}^\ast(\Gamma).
dbnvp Friday: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
8 Mar 3 Homework Assignment 6 (PDF)
dbnvp Monday: Lie algebraic weight systems.
dbnvp Wednesday: Graph cohomology and the construction of Z_0.
Graph Cohomology and Configuration Space Integrals (PDF)
dbnvp Friday: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
9 Mar 10 Homework Assignment 7 (PDF)
dbnvp Monday: The gl(N) weight system.
dbnvp Wednesday: The universal property, hidden faces.
dbnvp Friday: Insolubility of the quintic, naive expectations for CS perturbation theory.
10 Mar 17 Homework Assignment 8 (PDF)
dbnvp Monday: W_{\mathfrak g}\colon{\mathcal A}(\uparrow)\to{\mathcal U}({\mathfrak g}) and PBW.
dbnvp Wednesday: The anomaly.
dbnvp Friday: Faddeev-Popov, part I.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
11 Mar 24 Homework Assignment 9 (PDF)
dbnvp Monday: {\mathcal A} is a bi-algebra.
dbnvp Wednesday: Understanding and fixing the anomaly.
Friday: class cancelled.
12 Mar 31 Monday, Wednesday: class cancelled.
dbnvp Friday: A Monday class: back to expansions.
E Apr 7 dbnvp Monday: A Friday class on what we mostly didn't have time to do.
Add your name / see who's in!
Dror's Notebook
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0:27:16 [edit] A green knot.
0:43:23 [edit] (Note by User:Cameron.martin):

Claim: The number of legal 3-colorings of a knot diagram is always a power of 3.

This is an expansion on the proof given by Przytycki (https://arxiv.org/abs/math/0608172).

We'll show that the set of legal 3-colorings \mathcal{S} forms a subgroup of Z_3^r, for some r, which suffices to prove the claim. First, label each of the segments of the given diagram 1 through n, and denote a 3-coloring of this diagram by x = (x_1, x_2, ..., x_n), where each x_n is an element of the cyclic group of order 3 Z_3 = <a|a^3=1> (each element representing a different colour). It is clear that \mathcal{S} is a subset of Z_3^n. To show it is a subgroup, we'll take x = (x_1, x_2, ..., x_n), y = (y_1, y_2, ..., y_n) \in \mathcal{S}, and show that xy^{-1} = (x_1y_1^{-1}, x_2y_2^{-1}, ..., x_ny_n^{-1}) \in \mathcal{S}. It suffices to restrict our attention to one crossing in the given diagram, so we can without loss of generality let n = 3.

First, we (sub)claim that a crossing (involving colours x_1, x_2, x_3 is legal if and only if x_1x_2x_3 = 1 in Z_3. Indeed, if the crossing is legal, either it is the trivial crossing in which case their product is clearly 1, or each x_i is distinct, in which case x_1x_2x_3 = 1aa^2 = a^3 = 1. Conversely, suppose x_1x_2x_3 = 1, and suppose x_1 = x_2. It suffices to show that x_3 = x_1. This follows by case checking: if x_1 = 1, then 1 = x_1x_2x_3 = x_3; if x_1 = a, then 1=a^2x_3, implying that x_3 = a^{-2} = a; and if x_1 = a^2, then 1 = a^4x_3 = ax_3, implying that x_3 = a^{-1} = a^2. Thus, the subclaim is proven.

As a result, xy^{-1} = (x_1y_1^{-1}, x_2y_2^{-1}, x_3y_3^{-1}) satisfies x_1y_1^{-1}x_2y_2^{-1}x_3y_3^{-1} = (x_1x_2x_3)(y_3y_2y_1)^{-1} = 1 since both x, y \in \mathcal{S}. This implies that xy^{-1} \in \mathcal{S}, and hence shows that \mathcal{S} is a subgroup of Z_3^n for n = the number of line segments in the diagram. By Lagrange's theorem, the number of legal 3-colorings (the order of \mathcal{S}) is a power of 3.

(Note by User:Leo algknt):

Using linear Algebra: Idea from class on Wednesday 23 May, 2018

Let D be a knot diagram for the knot K with n crossings. There are n arcs. Let a_1, a_1, \ldots, a_n \in {\mathbb Z}_3 represent the arcs. Now let a,b,c \in {\mathbb Z}_3. Define \wedge : {\mathbb Z}_3 \times {\mathbb Z}_3 \rightarrow {\mathbb Z}_3 by

a\wedge b = 

a, &  a = b\\
c, & a\not= b
\right., so that a\wedge b + a + b \equiv 0\mod 3.

Then, with the above definition, we get a linear equation a_{i_1} + a_{i_2} + a_{i_3} \equiv 0\mod 3 for each each of the n crossings, where i_1, i_2, i_3 \in \{1, 2, \ldots, n\}. Thus we get a system of n linear equation, from which we get a matrix M. The nullspace \mathrm{Null}(M) of M is the solution to this system of equation and this is exactly the set of all 3-colourings of D. This is a vector space of size \lambda(K) =|\mathrm{Null}(M)| =  3^{\dim(\mathrm{Null}(M))}

0:46:29 [edit] (Note by User:Cameron.martin). The unknot $0_1$ and the figure-eight knot $4_1$ both have 3 legal 3-colorings, i.e. $\lambda(0_1) = \lambda(4_1) = 3$. 3-coloring fails to distinguish the unknot from the figure-eight. See http://katlas.math.toronto.edu/wiki/The_Rolfsen_Knot_Table for more information on specific knots.