Knot theory (190) course, Fall 2003

Monday-Wednesday-Friday from 2.00-2.50 in room HSS 2321.

Lecturer: Justin Roberts

The traditional first course in topology deals with ``point-set topology'': the study of metric and topological spaces, continuity, compactness, connectedness, and other properties beginning with ``c''.

This branch of the subject is really just a part of analysis, and while it is important for the foundations of the subject and can help you learn to write proofs properly, it can all seem very abstract and dry. Where are the doughnuts, coffee cups, pretzels, rubber sheets, knots and so on of popular topology?

Traditionally, we teach the more geometric, visual side of the subject after teaching all the basic tools. This is not unreasonable, but it does take a long time to do properly, and is quite hard to motivate because it turns history on its head. After all, people have been using and thinking about knots for thousands of years, but the definition of a topological space is only a hundred years old.

Fortunately it isn't necessary to work this way round. With a little care we can do quite a lot of knot theory without needing to talk about the foundational aspects of topology. I intend to start this way, and then try to develop a little bit of the abstract theory only if we really start to need it later on.

One of the common problems faced by students in topology is deciding how much detail to write in proofs; the subject spans a great range, from the most pedantic and precise point-set arguments, to visual arguments which can seem like ``hand-waving''. I hope that the course will help you in general to appreciate and produce ``good mathematics'' at whatever level is appropriate.

Knot theory

A closed loop of string in 3-space is called a knot. If it can be untangled (no cheating with scissors) into a planar circle it is called the unknot. The picture below shows the first few knots, in order of number of crossings; it starts with the unknot, then the trefoil and figure-eight:

To prove this we need to apply some algebraic topology (e.g. the fundamental group of the complement of the knot in space), geometric topology (e.g. looking at surfaces associated to the knot - we will spend some time on the topological classification of surfaces in its own right), combinatorial topology (e.g. counting 3-colourings of a diagram, or the famous Jones polynomial) or other cleverness. The study of knots is both a testbed in which to apply the abstract theory of topology, and a source of new problems and methods. Plus of course it's fun! (Challenge: make a trefoil as above, and then make one where all the overcrossings go under and vice versa. Are they the same?) The picture was grabbed from the wonderful website The KnotPlot Site; check it out!

Pre-requisites

To be honest, you don't need to know much to get started in this course! The most important thing is to be happy with linear algebra (e.g. Gaussian elimination), as we will use matrices occasionally. If you know anything about groups that might also help later on (though it depends how far we get).

Books

My intention is to use primarily the notes for the course I gave in Edinburgh, which are available by clicking here. (I hope to redevelop them a bit during the course.)

``Knots knotes'' in gzipped postscript.

``Knots knotes'' in pdf.

For a more complete reference though, including point-set topology and additional algebraic topology, I will also be using the following:

M. Armstrong, Basic Topology (1983, Springer-Verlag).

If you get really interested in knot theory and want a less mathematically-detailed but much broader survey of what is known (it's aimed at clever high-school students, but goes right up to current research problems!) you might want to have a look at the very readable:

C. Adams, The Knot Book (1994, W. H. Freeman)

Additional Information

Section The TA is Jana Comstock (jcomstoc@math.ucsd.edu). She will hold section twice on Mondays, at 12 in CSB 005 and at 3 in CSB 004. (Note: different from advertised time!) Her office hours are 2-4 Fridays in APM 2202.

My office hours are Thursday 5-6 and Friday 11-12 in room 7210 in APM.

Midterm will perhaps be on October 24th. (If anyone has any preferences I will bear them in mind. I don't mind when it is!)

Homeworks will be set most weeks on Mondays and be due the following Monday in the TA section. The list of problems will be maintained here: Homework problems

Grading will be 20% homework, 20% midterm, 60% final.

Further Topics in Topology (191) course, Winter 2004

There will be a follow-on course in the winter term. My current plan is to do the following topics:

1. Basic point-set topology (open and closed sets, etc.)

2. Basic algebraic topology (the fundamental group)

3. Topology of surfaces (basically as in the "knotes")

4. Geometry of surfaces (some hyperbolic geometry)

5. 3-manifolds (examples, surgery, discussion of current research)

I won't assume 190 as a prerequisite (though my expectation is that most of the class probably will have done it first.) I'm sure I will refer to examples from 190 occasionally, but as it has been entirely focussed on knot theory and combinatorics, nothing in 191 will really rely on it.

Hope to see you next term!