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.
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!
There are a couple of other useful references, but don't feel obliged
to buy these. If you want to know more about point-set topology and
the foundations of algebraic topology, look at
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)
My office hours are Wednesdays 11-12 and perhaps Fridays 4-5
in AP&M room 7210.
Midterm will be in class on Friday Oct 19, unless anyone has
any complaints!
Homeworks will be set most weeks on Fridays and be due the
following Thursday in section. The list of problems will be maintained
here: Homework problems
Grading will be 20% homework, 20% midterm, 60% final.
``Knots knotes'' in gzipped postscript.
``Knots
knotes'' in pdf.
Additional Information
Section The TA is Ben Hummon (bhummon@math.ucsd.edu). His
section is Thursday 8-9Warren 2114. His office hours are Tuesdays
2-4pm in APM 6446.