## Maths exercises

Here are some math exercises for the reader. They vary in difficulty.

(1)  Let x and y be non-commuting variables satisfying

xy = q yx

for some complex number q.
Write down a closed expression for (x+y)n, for all positive integers n. (Hint: your answer should resemble the Binomial theorem in the limit as q -> 1.)

(2)  Let x, y and z be non-commuting variables satisfying

xy = q yx + z,  xz = q2zxzy = q2zy,

for some complex number q.
Write down a closed expression for (x+y)n, for all positive integers n.

(3)  Create a complete link invariant, ie, an in-principle method of constructing a one-to-one correspondence between topological equivalence classes of links and some other collection of mathematical “things” (eg numbers).

(4)  Determine all the irreducible representations, and their tensor product theorems and “fusion rules”, for all quantum superalgebras at roots of unity.

I’ll give my very own “Clay Institute prize” of AUD \$10 to anyone who does all of these things.