## Contents

### Crucial Information

Agenda: Understand the promise and the difficulty of the not-yet-existant "Algebraic Knot Theory".

Instructor: Dror Bar-Natan, drorbn@math.toronto.edu, Bahen 6178, 416-946-5438. Office hours: by appointment.

Classes: Tuesdays 9-10 and Thursdays 10-12 at Bahen 6183.

SVN repository: http://katlas.math.toronto.edu/svn/06-1350/.

### Abstract

An "Algebraic Knot Theory" should consist of two ingredients

1. A map taking knots to algebraic entities; such a map may be useful, say, to tell different knots apart.
2. A collection of rules of the general nature of "if two knots are related in such and such a way, their corresponding algebraic entities are related in such and such a way". Such rules may allow us, say, to tell how far a knot is from the unknot or how far are two knots from each other.

(If you have seen homology as in algebraic topology, recall that its strength stems from it being a functor. Not merely it assigns groups to spaces, but further, if spaces are related by maps, the corresponding groups are related by a homomorphism. We seek the same, or similar, for knots.)

The first ingredient for an "Algebraic Knot Theory" exists in many ways and forms; these are the many types and theories of "knot invariants". There is very little of the second ingredient at present, though when properly generalized and interpreted, the so-called Kontsevich Integral seems to be it. But viewed from this angle, the Kontsevich Integral is remarkably poorly understood.

The purpose of this class will be to understand all of the above.

### Warning

This class is not for everyone. An old rule says one should not give a class on one's own current research. Here we will break that rule with vengeance - the class won't just be about current research, it will be about research that had not been done yet. Our purpose will not be to paint a beautiful picture of an established field, rather, to learn about the parts that may one day fit into and create such a beautiful picture, or may not. The parts are pretty in themselves and will force us to tour a number of deep mathematical fields. But by the nature of things, the presentation may well be confused and frustrating. If that scares you, or if all you need is a sure credit, do not take this class.

### Some Sub-Topics

The Jones polynomial; finite type invariants, universal finite type invariants; knotted trivalent graphs; tetrahedra, associators and all that; Lie algebras and weight systems; the envelopes of some classical invariants.

The stress of giving a coherent description of a non-existent subject will be too much for me. To mask this, whenever I will need a break we will branch off into asides, some more relevant and some less. Possible topics include: categorification, Chern-Simons and Feynman diagrams, Stonehenge pairings and configuration spaces, Lie algebras and the four-colour theorem, the Århus integral, multiple $\zeta$-numbers and more.

### Prerequisites

The graduate core classes in topology and in algebra, or anything equivalent to that.

### Wiki

The class web site is a wiki, as in Wikipedia - meaning that anyone can and is welcome to edit almost anything and in particular, students can post notes, comments, pictures, solution to open problems, whatever. Some rules, though -

• This wiki is a part of my (Dror's) academic web page. All postings on it must be class-related (or related to one of the other projects I'm involved with).
• Criticism is fine, but no insults or foul language, please.
• I (Dror) will allow myself to exercise editorial control, when necessary.
• The titles of all pages related to this class should begin with "06-1350/", just like the title of this page.

Some further editing help is available at Help:Contents.

### Good Deeds and The Final Grade

In addition, you will be able to earn up to 60 "good deeds" points throughout the year for doing services to the class as a whole. There is no pre-set system for awarding these points, but the following will definitely count:

• Solving an open problem.
• Giving a class on one subject or another.
• Drawing a beautiful picture to illustrate a point discussed in class and posting it on this site.
• Taking class notes in nice handwriting, scanning them and posting them here.
• Formatting somebody else's class notes, correcting them or expanding them in any way.
• Writing an essay on expanding on anything mentioned in class and posting it here; correcting or expanding somebody else's article.
• Doing anything on our 06-1350/To do list.
• Any other service to the class as a whole.

Good deed points will count towards your final grade! If you got $n$ of those, they are solidly your and the formula for the final grade below will only be applied to the remaining $100-n$ points. So if you got 40 good deed points (say) and your final grade is 80, I will report your grade as $40+80(100-40)/100=88$. Yet you can get an overall 100 even without doing a single good deed.

Important. For your good deeds to count, you must do them under your own name. So you must set up an account for yourself on this wiki and you must use it whenever you edit something. I will periodically check Recent changes to assign good deeds credits.

### Homework

There will be 4-5 problem sets. I encourage you to discuss the assignments with other students or even browse the web, so long as you do at least some of the thinking on your own and you write up your own solutions.

### Class Photo

To help me learn your names, I will take a class photo on Thursday of the third week of classes. I will post the picture on the class' web site and you will be required to send me an email and identify yourself in the picture or to identify yourself on the Class Photo page of this wiki.

### Old Title and Abstract

Before my decision to go radical and risky, the title of this class was "The Jones Polynomial" and the abstract read:

The Jones polynomial is perhaps the simplest knot invariant to define; it can be defined (and will be defined in the first class) in about 5 minutes, invariance can be proven in about 15 minutes, it can be programmed in another 10 minutes, and then it can be evaluated for the first few hundred knots in some 10 minutes or so. In the rest of the semester we will see that the Jones polynomial has some knot theoretic implications, has lovely generalizations and fits within some nice pictures, and is a wonderful excuse and unifying centre for the study of several other deep subjects, including but not limited to combinatorics, homological algebra, Lie algebras, quantum algebra, category theory and even quantum field theory. Some of these subjects we will cover in great detail; others, for the lack of time, will only be briefly touched. The prerequisites are the graduate core classes in topology and in algebra, or anything equivalent to that.

These old title and abstract are not entirely lies. I do plan to start with the Jones polynomial (it is quick fun), and in some historical sense, the Jones polynomial is the start of much of what we will do.