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

MAT1350F - Topics in Knot Theory

Toronto, Fall 2020

$$ $Essentials $$


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: $$make_page {name => 'About', title => 'About This Class'}$$.
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.
$$make_page {name => 'HW1', title => 'Homework Assignment 1'}$$.
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.
$$make_page {name => 'HW2', title => 'Homework Assignment 2'}$$.
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.
$$make_page {name => 'HW3', title => 'Homework Assignment 3'}$$.
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.
$$make_page {name => 'HW4', title => 'Homework Assignment 4'}$$.
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.
$$make_page {name => 'HW5', title => 'Homework Assignment 5'}$$.
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.
$$make_page {name => 'HW6', title => 'Homework Assignment 6'}$$.
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.
$$make_page {name => 'HW7', title => 'Homework Assignment 7'}$$.
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.
$$make_page {name => 'HW8', title => 'Homework Assignment 8'}$$.
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: