AKT-14/Tricolourability without Diagrams

DBN: Classes: 2013-14: AKT-14 / Navigation # Week of... Notes and Links 1 Jan 6 About This Class (PDF).  Monday: Course introduction, knots and Reidemeister moves, knot colourings. Tricolourability without Diagrams  Wednesday: The Gauss linking number combinatorially and as an integral.  Friday: The Schroedinger equation and path integrals. Friday Introduction (the quantum pendulum) 2 Jan 13 Homework Assignment 1.  Monday: The Kauffman bracket and the Jones polynomial.  Wednesday: Self-linking using swaddling.  Friday: Euler-Lagrange problems, Gaussian integration, volumes of spheres. 3 Jan 20 Homework Assignment 2.  Monday: The definition of finite-type and some examples.  Wednesday: The self-linking number and framings.  Friday: Integrating a polynomial times a Gaussian. Class Photo. 4 Jan 27 Homework Assignment 3.  Monday: Chord diagrams and weight systems.  Wednesday: Swaddling maps and framings, general configuration space integrals.  Friday: Some analysis of $d^{-1}$. 5 Feb 3 Homework Assignment 4.  Monday: 4T, the Fundamental Theorem and universal finite type invariants. The Fulton-MacPherson Compactification (PDF).  Wednesday: The Fulton-MacPherson Compactification, Part I.  Friday: More on pushforwards, $d^{-1}$, and $d^{\ast }$. 6 Feb 10 Homework Assignment 5.  Monday: The bracket-rise theorem and the invariance principle.  Wednesday: The Fulton-MacPherson Compactification, Part II.  Friday: Gauge fixing, the beginning of Feynman diagrams. R Feb 17 Reading Week. 7 Feb 24  Monday: A review of Lie algebras.  Wednesday: Graph cohomology and $\Omega _{dR}^{\ast }(\Gamma )$.  Friday: More on Feynman diagrams, beginning of gauge theory. From Gaussian Integration to Feynman Diagrams (PDF). 8 Mar 3 Homework Assignment 6 (PDF)  Monday: Lie algebraic weight systems.  Wednesday: Graph cohomology and the construction of $Z_{0}$. Graph Cohomology and Configuration Space Integrals (PDF)  Friday: Gauge invariance, Chern-Simons, holonomies. Mar 9 is the last day to drop this class. 9 Mar 10 Homework Assignment 7 (PDF)  Monday: The $gl(N)$ weight system.  Wednesday: The universal property, hidden faces.  Friday: Insolubility of the quintic, naive expectations for CS perturbation theory. 10 Mar 17 Homework Assignment 8 (PDF)  Monday: $W_{\mathfrak {g}}\colon {\mathcal {A}}(\uparrow )\to {\mathcal {U}}({\mathfrak {g}})$ and PBW.  Wednesday: The anomaly.  Friday: Faddeev-Popov, part I. Gaussian Integration, Determinants, Feynman Diagrams (PDF). 11 Mar 24 Homework Assignment 9 (PDF)  Monday: ${\mathcal {A}}$ is a bi-algebra.  Wednesday: Understanding and fixing the anomaly. Friday: class cancelled. 12 Mar 31 Monday, Wednesday: class cancelled.  Friday: A Monday class: back to expansions. E Apr 7  Monday: A Friday class on what we mostly didn't have time to do. Add your name / see who's in! Dror's Notebook

Here are some thoughts on how to define tricolourability without choosing a diagram.

Another place in which arcs of a diagram come up naturally is in the Wirtinger presentation for the fundamental group of the knot complement. Here is how the presentation is defined: each arc of the knot diagram corresponds with a generator, and each crossing corresponds with a relation between the generators of the incident arcs of the form $xy=yz$ where $y$ is the generator corresponding with the overcrossing.

Now if $\langle S\mid {\text{relations}}\rangle$ is a Wirtinger presentation for a knot diagram, it's natural to think of a tricolouring as a map $\phi :S\rightarrow \{R,G,B\}$. We'd like to try to extend this to a group homomorphism $\phi :\langle S\mid {\text{relations}}\rangle \rightarrow \langle R,G,B\mid {\text{relations}}\rangle$. This works if target group has the relation $RG=GB$ along with all other relations obtained by permuting $R,G,B$. These relations fix the target group as $D_{2\cdot 3}$.

Thus, we've associated with each tricolouring a homomorphism from the fundamental group of the knot complement to $D_{2\cdot 3}$. Not every such homomorphism gives a tricolouring; for example, take the trivial homomorphism. I believe that the following is a sufficient condition for a homomorphism $\phi$ to give a tricolouring: for every element $x$ in $\pi _{1}(\mathbb {R} ^{3}\setminus K)$ whose representative as a loop in $\mathbb {R} ^{3}\setminus K$ has odd linking number with $K$, $\phi (x)$ is an order 2 element in $D_{2\cdot 3}$. Hence, we can define tricolourings as certain kinds of homomorphisms from $\pi _{1}(\mathbb {R} ^{3}\setminus K)$ to $D_{2\cdot 3}$ without having to choose a diagram.