\( \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calT{{\mathcal T}} \def\Lim{{\operatorname{Lim}}} \) © | Dror Bar-Natan: Classes: 2021-22:

MAT1350F - Algebraic Knot Theory and Computation

Toronto, Fall 2021

Disclaimer. Everything on this page remains tentative until it had happenned. The Evil Virus situation may force us to make last minute changes to anything.

Agenda. Find a poly-time-computable strong knot invariant with good algebraic properties.

Abstract. The destination will be "a poly-time computable strong knot invariant with good algebraic properties". But you will be taking the course for the journey, not for the destination: What are knots and what are some of the problems around them? Why care about "invariants with good algebraic properties"? What is the "Yang-Baxter equation"? What are "virtual tangles"? What are "Hopf algebras"? Why would a topologist care about computations in Heisenberg algebras more than most physicists? How does Gaussian integration, and how do Feynman diagrams, arise in pure algebra? What is the "Drinfel'd Double Procedure"? Are we there yet?

The professor for this class does not believe anything that he does unless it is coded and the code runs. A useful life skill you will learn here is that even the incredibly abstract can become a computer program, often with no loss to its beauty.

Even more details are here.

Instructor. Dror Bar-Natan, drorbn@math.toronto.edu (for course administration matters only; math on email is slow and prone to misunderstandings, so I generally avoid it). Office: Bahen 6178.

Classes. Mondays and Wednesdays at 3-4PM and Fridays at 1-2PM, at Bahen 6178. A complete video record will be available here, at a few hours delay.

Office Hours. Tuesdays at 9:30-10:30 at Bahen 6178 and online at http://drorbn.net/vchat.

Texts. No "course text"! But maybe one day I will write a text following this course...

Course Calendar

# Week of ...  
1 September 6 UofT graduate classes begin on Thursday September 9 and our first class is on Friday September 10.
Handout: Some Non Obvious Examples.
Handout: A Knot Zoo.
Handout: Tentative Plan.
Handout: About This Class.
Friday: Course Introduction. Videos: @dbnvp, @YouTube.
2 September 13 Monday: What are knots? And a word about the Kauffman bracket. Videos: @dbnvp, @YouTube.
Wednesday: The Jones polynomial and implementation. Videos: @dbnvp, @YouTube. Mathematica session: pdf, nb.
Homework Assignment 1.
Friday: A faster Jones program. Videos: @dbnvp, @YouTube. Mathematica session 1: pdf, nb. Mathematica session 2: pdf, nb.
3 September 20 Monday: Tangles and planar algebras. Videos: @dbnvp, @YouTube.
Link: On Beyond Zebra!.
Wednesday: Genus, the ribbon property. Videos: @dbnvp, @YouTube.
Handout: Knots and Surfaces.
Friday: Surfaces and the Seifert algorithm. Videos: @dbnvp, @YouTube.
Homework Assignment 2.
4 September 27 Monday: Tangles, genus, ribbon. Videos: @dbnvp, @YouTube.
Wednesday: The Yang-Baxter technique. Videos: @dbnvp, @YouTube.
Friday: IHOP Algebras, $R$-Elements, and $WG$. Videos: @dbnvp, @YouTube. Mathematica session: pdf, nb.
Handout: IHOP Algebras, $R$-Elements, and $WG$.
Homework Assignment 3.
5 October 4 Monday: Computations with WG. Videos: @dbnvp, @YouTube. Mathematica session: pdf, nb.
Wednesday: Meta-monoids and virtual tangles. Videos: @dbnvp, @YouTube.
Friday: Rotational Virtual Knots (1). Videos: @dbnvp, @YouTube.
Homework Assignment 4.
6 October 11 Monday is Thanksgiving; no class.
Wednesday: The Kerler Algebra. Videos: @dbnvp, @YouTube. Mathematica session: pdf, nb.
Friday: The 4x4 Alexander R-Matrix; conversions to RVK. Videos: @dbnvp, @YouTube. Mathematica session: pdf, nb.
7 October 18 Monday: More computations, a bit on the Heisenberg algebra. Videos: @dbnvp, @YouTube. Mathematica session: Common@.nb - pdf, nb and 4DAlexander2@.nb - pdf, nb.
Wednesday: More on the Heisenberg algebra and a bit on generating functions. Videos: @dbnvp, @YouTube.
Friday: The generating function of the Heisenberg multiplication. Videos: @dbnvp, @YouTube.
8 October 25 Monday: Composing generating functions. Videos: @dbnvp, @YouTube.
Wednesday: Composing Gaussians. Videos: @dbnvp, @YouTube. Mathematica session: GDO-Heisenberg@.pdf, GDO-Heisenberg@.nb.
Friday: Implementing GDO. Videos: @dbnvp, @YouTube. Mathematica session: GDO-Heisenberg2@.pdf, GDO-Heisenberg2@.nb.
9 November 1 Monday: Contracting Gaussians: Algebra by the means of partial differential equations. Videos: @dbnvp, @YouTube.
Wednesday: Contracting Gaussians (2). Videos: @dbnvp, @YouTube.
Friday: Gamma Calculus. Videos: @dbnvp, @YouTube.
R November 8 Reading Week, no classes.
10 November 15 Monday: Perturbed Gaussian Integration. Videos: @dbnvp, @YouTube.
Wednesday: Feynman Diagrams. Videos: @dbnvp, @YouTube.
Friday: Philosophical review. Videos: @dbnvp, @YouTube.
11 November 22 Monday: Lemmas 1 and 2. Videos: @dbnvp, @YouTube.
Wednesday: Perturbations and PDEs. Videos: @dbnvp, @YouTube. Mathematica session: PerturbedHeisenberg.pdf, PerturbedHeisenberg.nb.
Friday: Computations with perturbed Heisenberg. Videos: @dbnvp, @YouTube. Mathematica session: PerturbedHeisenberg2@.pdf, PerturbedHeisenberg2@.nb.
12 November 29 Monday: CU and QU. Videos: @dbnvp, @YouTube.
Handout: CU and QU.
Homework Assignment 5.
Wednesday: Multiplication in CU. Videos: @dbnvp, @YouTube.
Friday: Mostly DPG2. Videos: @dbnvp, @YouTube.
13 December 6 Monday: Primitive and Group Like Elements. Videos: @dbnvp, @YouTube.
Wednesday: Discussion; not recorded.

Further resources: