\( \def\bbN{{\mathbb N}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calT{{\mathcal T}} \def\Lim{{\operatorname{Lim}}} \) © | Dror Bar-Natan: Classes: 2020-21:

MAT1350F - Topics in Knot Theory

Toronto, Fall 2020

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. Multiple tastes of multi-faceted Knot Theory.

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 (these days, mostly vacant).

Classes. Mondays, Wednesdays and Fridays 1-2PM, online at http://drorbn.net/vclass.

Office Hours. Tuesdays ay 9-10AM and at 1-2PM, online at http://drorbn.net/vchat.

Texts. No "course text", yet we will use material from the books by Lickorish, Burde and Zieschang, Kauffman, Rolfsen, and more.

Current Class Notes. Are here.

Course Calendar

# Week of ...  
1 September 7 UofT graduate classes begin on Tuesday September 8 and our first class is on Wednesday September 9.
Handout: About This Class.
Wednesday: Knots, knot diagrams, Reidemeister moves, 3-colourings. Videos: @dbnvp, @YouTube.
Friday: The Kauffman Bracket. Videos: @dbnvp, @YouTube.
Class notes for week 1.
2 September 14 Monday: The Jones polynomial and implementation. Videos: @dbnvp, @YouTube. Mathematica session: pdf, nb.
Wednesday: Faster implementations of the bracket. Videos: @dbnvp, @YouTube. Mathematica session: pdf, nb.
Homework Assignment 1.
Friday: Tangles and planar algebras. Videos: @dbnvp, @YouTube.
Class notes for week 2.
3 September 21 Monday: Last day to add this class.
Monday: End of the Jones story, beginning of Khovanov's. Videos: @dbnvp, @YouTube.
Homework Assignment 2.
Wednesday: Complexes, heights, degrees. Videos: @dbnvp, @YouTube.
Wednesday: HW1 was due at 11:59PM.
Friday: Constructing the Khovanov complex. Videos: @dbnvp, @YouTube.
Class notes for week 3.
4 September 28 Monday: Products, co-products, the Frobenius condition. Videos: @dbnvp, @YouTube.
Homework Assignment 3.
Wednesday: Proof of invariance of Khovanov homology. Videos: @dbnvp, @YouTube.
Wednesday: HW2 due at 11:59PM.
Friday: Some computations of Khovanov homology; finite type invariants. Videos: @dbnvp, @YouTube.
Class notes for week 4.
5 October 5 Monday: Examples of finite-type invariants. Videos: @dbnvp, @YouTube.
Wednesday: Chord diagrams, weight systems, FI, 4T. Videos: @dbnvp, @YouTube.
Homework Assignment 4.
Wednesday: HW3 due at 11:59PM.
Friday: The Algebra ${\mathcal A}$. Videos: @dbnvp, @YouTube.
Class notes for week 5.
6 October 12 Monday is Thanksgiving; no class.
Wednesday: $W_C$ and $\Box$. Videos: @dbnvp, @YouTube.
Current class notes.
Homework Assignment 5.
Thursday: HW4 due at 11:59PM.
Friday: $\calA$ and $\calA^t$. Videos: @dbnvp, @YouTube.
Class notes for week 6.
7 October 19 Monday: $\calA$ and $\calA^t$, Lie algebras. Videos: @dbnvp, @YouTube.
Wednesday: Lie algebraic weight systems. Videos: @dbnvp, @YouTube.
Homework Assignment 6.
Wednesday: HW5 was due at 11:59PM.
Friday: $gl(N)$ and $sl(2)$. Videos: @dbnvp, @YouTube.
Class notes for week 7.
8 October 26 Monday: The Four Colour Theorem, Universal Finite Type Invariants. Videos: @dbnvp, @YouTube.
Monday: Last day to drop this class.
Wednesday: On HW6, on cosmic coincidences. Videos: @dbnvp, @YouTube.
Friday: Cosmic coincidences and IHX. Videos: @dbnvp, @YouTube.
Class notes for week 8.
Friday: HW6 is due at 11:59PM.
9 November 2 Monday: A quick and vague introduction to Feynman diagrams. Videos: @dbnvp, @YouTube.
Wednesday: The Knot Group. Videos: @dbnvp, @YouTube.
Homework Assignment 7.
Friday: The power of the Knot Group. Videos: @dbnvp, @YouTube.
Class notes for week 9.
R November 9 Undergraduate Reading Week - no classes.
10 November 16 Monday: Knot colourings and quandles. Videos: @dbnvp, @YouTube.
Wednesday: Mostly computer trouble, plus little more about quandles and little about Seifert surfaces. Videos: @dbnvp, @YouTube.
Wednesday: HW7 due.
Homework Assignment 8.
Friday: Seifert surfaces and the genus. Videos: @dbnvp, @YouTube.
Class notes for week 10.
11 November 23 Monday: Proof of the additivity of the genus. Videos: @dbnvp, @YouTube.
Wednesday: Continuous pathologies in R^3, and some smooth theorems. Videos: @dbnvp, @YouTube.
Thursday: HW9 is on Crowdmark and here.
Thursday: HW8 due.
Friday: Proof of unique factorization. Videos: @dbnvp, @YouTube.
Class notes for week 11.
12 November 30 Monday: Braids, closures, and links. Videos: @dbnvp, @YouTube.
Wednesday: Braids and complexity, pure braids, semi-direct products. Videos: @dbnvp, @YouTube.
Thursday: HW10 is on Crowdmark and here.
Thursday: HW9 was due.
Friday: Combing braids. PvB and PwB. Videos: @dbnvp, @YouTube.
Class notes for week 12.
13 December 7 Monday: More on w-braids and v-knots. Videos: @dbnvp, @YouTube.
Lickorish on unique factorization.
Wednesday: Expansions for groups. Videos: @dbnvp, @YouTube.
Class notes for week 13.
Wednesday: HW10 due at 11:59PM.

Further resources: