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### 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: Tuesday 5-8PM at Bahen 6183.

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

Get-Togethers: I (Dror) will not organize after-class get-togethers for food, drink, ping pong, conversation or anything, neither in nor out of Bahen. But if somebody will organize and invite, I will often join.

### Abstract

This is a continuation of 06-1350 and the abstract remains the same:

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.

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, more on Chern-Simons and Feynman diagrams, more on Stonehenge pairings and configuration spaces, the Århus integral, multiple ${\displaystyle \zeta }$-numbers and more.

### Prerequisites

Having taken 06-1350 or having made a heroic effort to catch up.

### 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 "07-1352/", just like the title of this page.

Some further editing help is available at Help:Contents.

### Good Deeds and The Final Grade

The grading scheme will be announced a few weeks into the class. Whatever it will be, it will allow you to earn some "good deeds" points throughout the semester 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 07-1352/To do list.
• Any other service to the class as a whole.

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 0-3 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.