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# AKT-140120 Video

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Notes on AKT-140120:    [edit, refresh]

The definition of finite-type and some examples.

# Week of... Notes and Links
: 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.
Friday Introduction (the quantum pendulum)
2 Jan 13 Homework Assignment 1.
: The Kauffman bracket and the Jones polynomial.
: 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.
Class Photo.
4 Jan 27 Homework Assignment 3.
: Chord diagrams and weight systems.
: Swaddling maps and framings, general configuration space integrals.
: Some analysis of $d^{-1}$.
5 Feb 3 Homework Assignment 4.
: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
: The Fulton-MacPherson Compactification, Part I.
: More on pushforwards, $d^{-1}$, and $d^\ast$.
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.
7 Feb 24 : A review of Lie algebras.
: Graph cohomology and $\Omega_{dR}^\ast(\Gamma)$.
: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
8 Mar 3 Homework Assignment 6 (PDF)
: Lie algebraic weight systems.
: Graph cohomology and the construction of $Z_0$.
Graph Cohomology and Configuration Space Integrals (PDF)
: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
9 Mar 10 Homework Assignment 7 (PDF)
: The $gl(N)$ weight system.
: The universal property, hidden faces.
: Insolubility of the quintic, naive expectations for CS perturbation theory.
10 Mar 17 Homework Assignment 8 (PDF)
: $W_{\mathfrak g}\colon{\mathcal A}(\uparrow)\to{\mathcal U}({\mathfrak g})$ and PBW.
: The anomaly.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
11 Mar 24 Homework Assignment 9 (PDF)
: ${\mathcal A}$ 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.

Dror's Notebook
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0:00:33 [add] Reminder on Kauffman and Jones.
0:01:22 [add] Notices and the proper spelling on Wensday.
0:15:54 [add] $n$-singular knots and differentiating invariants.
0:22:10 [add] The definition of finite-type.
0:22:11  Much like the concept of a type n knot invariant described in this lecture, there is such a thing as a type n invariant of "virtual knots". Virtual knots can be defined (among other ways) as equivalence classes of Gauss diagrams under a certain set of generalized Reidemeister moves. Refer to the note at time 31:27 of the video found at http://drorbn.net/dbnvp/AKT-140127.php, and Goussarev, Polyak, and Viro's paper (https://arxiv.org/abs/math/9810073) for more information on Gauss diagrams and the specific set of moves. Equivalently, one can think of virtual knots as quadrivalent planar graphs, where each vertex can either be the usual over/undercrossing pair as in regular knots, or a so-called "virtual crossing", at which the lines simply cross each other with no extra information. Analogous to the double point in the regular knot situation is the "semi-virtual crossing", given by an analogous relation (illustrated below).

An invariant $v: \mathcal{VK} \rightarrow A$ (where $\mathcal{VK}$ is the space of virtual knots and $A$ is some abelian group) of virtual knots is said to be of type $n$ if it vanishes on virtual knots with $n+1$ semi-virtual crossings. By virtue of equation (3) above, it can be shown that type $n$ virtual knot invariants, when restricted to honest knots, are knot invariants of type at least n. Namely, a type n virtual knot invariant evaluated on $n+1$ double points is a sum of evaluations of $v$ on $n+1$ semi-virtual crossings (by equation 3), and is therefore equal to 0.

0:30:43 [add] Jones is a FT series.
0:34:12  The second crossing on this line should be an undercrossing (as stated), not an overcrossing (as drawn). Namely, in the blackboard shot below the third line from the top should be $q^{-1}J(+)-qJ(-)=\ldots$, and not as written.
0:35:15  The Jones Skein relation is quite similar to the skein relation for the Alexander polynomial.
0:50:12 [add] The Jones skein relation.
0:52:10 [add] Jones is a FT series, proof.