Knot theory (190) course, Fall 2010

Monday-Wednesday-Friday from 3.00-3.50 in AP&M B412.

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?) 

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 my own notes. These were originally written for a course I gave in Edinburgh, and they don't correspond exactly to how I will teach the course this year. But I will be rewriting parts of them during the term, I hope. To start with, I would recommend that if you print them, you only print the first 4 sections! It is the latter parts (5 and onwards) which will be most heavily altered, and I hope that I will have this done by the time we need them.


``Knots knotes'' (draft version Sept 24, 2010) in pdf.

Here are some other useful references (but don't feel obliged to buy any of them!) I've put these in order of "level of mathematical sophistication" - easiest first.

C. Adams, The Knot Book (1994, W. H. Freeman) This book is a survey of knot theory. It isn't a typical textbook - it is not very detailed mathematically, since it's actually aimed at clever high-school students! But it does attempt to give the flavour of some really quite advanced topics, including current research and open problems!)

M. Armstrong, Basic Topology (1983, Springer-Verlag). This is a nice undergrad-level book which teaches point-set topology and the foundations of algebraic topology. It's not directly relevant to the course, but if you are interested in point-set topology, this is probably the best place to look.

N. Gilbert and T.Porter, Knots and Surfaces (1994, OUP) An undergrad-level book, which as I remember contains some basic point-set topology too. Its focus is extremely algebraic, however - it goes into group-theoretic aspects in a lot of detail.

A. Sossinsky, Knots, (2004, Harvard). This is a translation of a French edition, "Noeuds". It is quite idiosyncratic, and has a lot of typos and mistakes, so I don't much recommend it. However, you simply must check out the awesome picture of a particular type of eel which knots itself in order to stay slippery!

C. Livingston, Knot Theory (1993, Carus Mathematical Monographs) A nice small book which surveys knot theory, again without pedantic detail, but from a perspective of someone who knows point-set topology and basic algebraic topology. So it's more a grad-level book, but it's very nicely written and perhaps still useful.

V. Manturov, Knot Theory, (2004, CRC) This book is quite new, and contains the most up-to-date results of any of the books listed here. I haven't done more than skim it myself though. I think it's a grad-level book really. Marcus Amezcua pointed out that there is a free online copy available here.

D. Rolfsen, Knots and Links (1976, Publish or Perish) The "old testament" of knot theory, with a picture on every page. It is a serious graduate-level book and it relegates a lot of proofs to exercises, but it is a unique and classic book, which you might simply enjoy paging through. (It was written before the Jones polynomial was discovered.)

W. B. R. Lickorish, An Introduction to Knot Theory (1997, Springer GTM) The "new testament" of knot theory, a graduate-level textbook dealing especially with post-Jones-polynomial knot theory. It's by my PhD advisor - you might enjoy his dry wit!

Websites

There are lots of knot theory resources on the web these days - these are the main ones that spring to mind. You'll learn a lot just by surfing these sites, and they also provide tables of calculations of pretty much every knot invariant you can think of, some software for drawing and calculating with knots, and beautiful images.

The Knot Atlas (wiki)

The KnotPlot Site

Table of Knot Invariants

Additional Information

Section The TA is Lyla Fadali (lfadali@math.ucsd.edu). Sections are 3-4 and 4-5 on Thursday in B412. Her office hours are TBA in APM 6341.

My office hours are Mon 11-12 and Wed 4-5 in AP&M room 7210.

Midterm will be in class on Friday Oct 15, unless anyone has any complaints!

Homeworks will be set most weeks on Fridays and be due by 3pm the following Friday. The list of problems will be maintained here: Homework problems

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