|© | Dror Bar-Natan: Classes: 2013-14: AKT:||< >|
|width: 400 720||ogg/AKT-140106_400.ogg||orig/AKT-140106.MOD|
|Videography by Iva Halacheva||troubleshooting|
Course introduction, knots and Reidemeister moves, knot colourings.
|#||Week of...||Notes and Links|
|1||Jan 6|| (PDF). |
: Course introduction, knots and Reidemeister moves, knot colourings.
Tricolourability without Diagrams
: The Gauss linking number combinatorially and as an integral.
: The Schroedinger equation and path integrals.
|2||Jan 13||Homework Assignment 1. |
: The Kauffman bracket and the Jones polynomial.
: Self-linking using swaddling.
: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
|3||Jan 20||Homework Assignment 2. |
: The definition of finite-type and some examples.
: The self-linking number and framings.
: Integrating a polynomial times a Gaussian.
|4||Jan 27||Homework Assignment 3. |
: Chord diagrams and weight systems.
: Swaddling maps and framings, general configuration space integrals.
: Some analysis of .
|5||Feb 3||Homework Assignment 4. |
: 4T, the Fundamental Theorem and universal finite type invariants.
: The Fulton-MacPherson Compactification, Part I.
: More on pushforwards, , and .
|6||Feb 10||Homework Assignment 5. |
: The bracket-rise theorem and the invariance principle.
: The Fulton-MacPherson Compactification, Part II.
: Gauge fixing, the beginning of Feynman diagrams.
|R||Feb 17||Reading Week.|
|7||Feb 24||: A review of Lie algebras. |
: Graph cohomology and .
: More on Feynman diagrams, beginning of gauge theory.
|8||Mar 3|| (PDF) |
: Lie algebraic weight systems.
: Graph cohomology and the construction of .
: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
|9||Mar 10|| (PDF) |
: The weight system.
: The universal property, hidden faces.
: Insolubility of the quintic, naive expectations for CS perturbation theory.
|10||Mar 17|| (PDF) |
: and PBW.
: The anomaly.
: Faddeev-Popov, part I.
|11||Mar 24|| (PDF) |
: is a bi-algebra.
: Understanding and fixing the anomaly.
Friday: class cancelled.
|12||Mar 31||Monday, Wednesday: class cancelled. |
: A Monday class: back to expansions.
|E||Apr 7||: A Friday class on what we mostly didn't have time to do.|
Add your name / see who's in!
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 forms a subgroup of , 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 , where each is an element of the cyclic group of order 3 (each element representing a different colour). It is clear that is a subset of . To show it is a subgroup, we'll take , and show that . 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 is legal if and only if in . Indeed, if the crossing is legal, either it is the trivial crossing in which case their product is clearly 1, or each is distinct, in which case . Conversely, suppose , and suppose . It suffices to show that . This follows by case checking: if , then ; if , then , implying that ; and if , then , implying that . Thus, the subclaim is proven.
As a result, satisfies since both . This implies that , and hence shows that is a subgroup of for n = the number of line segments in the diagram. By Lagrange's theorem, the number of legal 3-colorings (the order of ) is a power of 3.
(Note by User:Leo algknt):
Using linear Algebra: Idea from class on Wednesday 23 May, 2018
Let be a knot diagram for the knot with crossings. There are arcs. Let represent the arcs. Now let . Define by
, so that .
Then, with the above definition, we get a linear equation for each each of the crossings, where . Thus we get a system of linear equation, from which we get a matrix . The nullspace of is the solution to this system of equation and this is exactly the set of all 3-colourings of . This is a vector space of size