These problems are not generally like calculus problems, or even linear algebra ones. Instead of doing a lot of calculations with formulae, you will often find that you have to present a logical argument or explanation in English. Please try to write coherently! One helpful tip is to keep the reader in mind: instead of just jotting down your thoughts, imagine that you are actually writing to a fellow member of the class and that they will argue with you if you write confusing things!
Homework 1, Oct. 1; due Oct. 6
From the "knotes" do exercises 1.2.6, 1.2.9, 1.2.10, 1.6.2, 1.6.7
Also do this additional problem:
1A: Imagine you wanted to write a computer program (an algorithm) to classify the different equivalence classes of knots, and produce a table like the one I gave out. Which bits are going to be the easy parts, and which will be the hardest parts? (The idea is to try to get to grips with the fact that a computer can only perform tasks that will take a finite amount of time and memory.)
Homework 2, Oct. 8; due Oct. 13
From the "knotes" do exercises 2.3.3, 3.2.5, 3.2.6
Also do these additional problems:
2A: Draw a picture to show that there exist oriented 2-component links with arbitrary (integer) linking number
2B: Suppose L is an oriented link whose components are numbered 1 to n. We can define a "pairwise linking number" Lij by deleting all but the components numbered i and j and then computing their linking number in the usual way. Show that Lij is an invariant of L, and that the linking number of L is the sum of the various Lij.
2C: Show that any (oriented) Brunnian link with more than two components has linking number zero.
Homework 3, Oct. 13; due Oct. 20
From the "knotes" do exercises 3.3.5, 3.3.8, 3.3.10.
Also do these additional problems:
3A: Calculate the number of 5-colourings of the knot 61.
3B: Calculate the numbers of 5-colourings and 7-colourings of the knot
71.
MIDTERM INFORMATION
The midterm will be on Friday 24th in class. (Please bring paper or a
blue book.) It will cover everything we've done so far on knots,
diagrams, invariants and colourings, but not the Jones polynomial
stuff.
Here's the midterm from two years ago, for comparison,
in ps or
pdf. The
missing diagrams were, I think, the granny and reef knots from page 19
of the "knotes"!
Homework 4, Nov. 3; due Nov. 10
From the "knotes" do exercises 4.4.9, 4.4.10, 4.4.11, then the
following:
A. Show that the Jones polynomial of any knot, evaluated when t=1, is
equal to 1.
B: Suppose a diagram D contains a subdiagram R which has exactly four
strings emerging from it, as shown below. (For example, R might be a
single crossing, or one of the diagrams used in Reidemeister II, or
much more complicated.) Let D' be the new diagram obtained by rotating
the part R by 180 degrees. This operation is called mutation
of the diagram. Show that the Kauffman brackets of D and its
mutant D' are equal.
C: Show that the number of regions in a diagram of a knot equals the
number of crossings plus 2. (It is quite tricky to come up with a good
argument for this. One possibility is to just try to see how the
numbers change during the process of drawing the curve with a
continuous motion of the pen.)
Homework 5, Nov. 12; due Nov. 17
(Sorry for the delay in announcing this.) From the "knotes" do
exercises 4.4.14, 15, 16, 17.
SPECIAL ANNOUNCEMENT
Here is a link to a webpage I have with pictures of wallpaper
patterns, and listing some interesting pages about them. You might
find it fun.
Wallpaper
Homework 6, Nov. 24; due Dec. 1
A: Consider the identity braid in the group Bn. Now rotate
the top end of the braid through 360 degrees anticlockwise (imagine
that the strings are attached to a metal plate and that you rotate the
whole plate to do this). What you get is an interesting braid
Tn, called the full twist. (a) Find an
expression for the full twist in terms of the braid group's
generators. (It helps to draw explicit pictures for small values of n
first). (b) Show that Tn commutes with every element of
Bn. (There are two ways to do this: one entirely
pictorial, and one algebraic.)
B: (Fun for all the family, ages 6-66...)The homophony group
(of English) is the group with 26 generators a,b, ..., z and
one relation for every pair of English words which sound the
same. Example: "knight =
night" shows that k=1, "earn=urn" shows that
ea=u, and so on. (Scrabble rules apply; no proper names or
slang!) Prove that the group is trivial! (I would guess it's also
trivial in French but probably non-trivial in Spanish and very
non-trivial in Japanese!).
Note: the final exam is on Wed Dec 10 at 3pm in the usual room HSS
2321. It will consist of questions concentrating on the first part of
the course (knots, colourings, the Jones polynomial) rather than the
material on groups, though there may be something on this. It's hard
to think of questions so I can't stretch to making a "mock" exam, but
the midterm is a reasonable guide to the style. The final may be a bit
longer, that's all.
Office hours: I have office hours Monday Dec 1 5.30-6.30, also the
usual Thursday and Friday times, then next Monday at 2pm (or by
appointment).