© | Dror Bar-Natan: Classes: 2013-14: AKT:  < > 

AKT-140108 Video

width: 400 720 download ogg/AKT-140108_400.ogg
Videography by Iva Halacheva troubleshooting

Notes on AKT-140108:    [edit, refresh]

The Gauss linking number combinatorially and as an integral.


[edit navigation bar]

# 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.
AKT-14-ClassPhoto.jpg
Add your name / see who's in!
Dror's Notebook
AKT-14-Splash.png
refresh
panel
Managed by dbnvp: The small-font numbers on the top left of the videos indicate the available native resolutions. Click to test.

0:08:12 [edit] 18S-AKT Question: Why is this sum divisible by 2? Why the $\frac{1}{2}$?

Attempt 1 The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.

Jordan Curve Theorem. If $C$ is a simple closed curve in $\mathbb{R}^2$, then the complement ${\mathbb R}^2\setminus C$ has two components, the interior and the exterior, with $C$ the boundary of each.


Attempt 2The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even number of $\pm 1$'s in the computation of $lk(L)$, which yields an even number. It is always an integer. This is why we have a factor of $\frac12$.


Attempt 3 Improved The Jordan curve theorem requires that we have simple closed curves and in general the components of a link are not simple closed curves. However, in the definition of the linking number, only the crossings between the components count. So we can smooth out crossings that are not between the components and this will not affect the linking number; we will then get components that are simple closed curves. From this argument, we can then apply the Jordan curve theorem to get the desired result.

0:10:56 [add] The linking number as a sum over xings.
0:18:57 [add] Invariance of the linking number sum.
0:23:10 [add] The linking number as an integral.
0:24:50 [edit] I am still not clear on why the image of the unlink is the equator of the sphere. I am wandering why not any other circle above or below the equator. Do we homotop the image to the equator?

Since we have assumed that the embedding is invariant, we can keep the two circles in the same plane. Then, the directional vector would always be inside of the plane. Since we are free to choose which plane we want the unlink circles to be, without loss of generality, the two circles are chosen to be in the xy-plane. Then the directional vector will not have the z-component, hence the image of the map can only be on the equator (if we choose some other planes, the image would be a great circle on S^2 by rotating the equator).

0:34:01 [add] Computing the linking number integral.
0:36:25 [add] Invariance of the linking number integral.
0:42:40 [add] Alternative choice of volume forms.
0:49:08 [add] An ugly explicit formula for the linking number integral.