# 06-1350/Homework Assignment 1

1. Let ${\displaystyle p}$ be an odd prime. A knot diagram ${\displaystyle D}$ is called ${\displaystyle p}$-colourable if there is a non-constant map ("colouring") from the arcs of ${\displaystyle D}$ to ${\displaystyle {\mathbb {Z} }/p}$ so that at every crossing, the average of the colours of the two "under" arcs is equal to the colour of the "over" arc (calculations in ${\displaystyle {\mathbb {Z} }/p}$, of course).
2. Show that "being ${\displaystyle p}$-colourable" is invariant under Reidemeister moves and hence defines a knot invariant.
3. (Hard and not mandatory) Prove that the (5,3) torus knot T(5,3) (pictured above) is not ${\displaystyle p}$-colourable for any ${\displaystyle p}$.
2. Use the recursion formula $\displaystyle q^{-1}J(\overcrossing)-qJ(\undercrossing)=(q^{1/2} - q^{-1/2})J(\smoothing)$ and the initial condition ${\displaystyle J(\bigcirc )=1}$ to compute the Jones polynomial $\displaystyle J(\HopfLink)$ of the Hopf link and the Jones polynomial $\displaystyle J(\righttrefoil)$ of the right handed trefoil knot.