Crossing number and Torus links

Question

We define the crossing number of a knot $K$ to be the minimal number of crossings in any diagram of $K$. Surely we can easy prove that there do not exist knots with crossing number $1$ and $2$ (because suppose well, then such knots are equivalent with the unknot, thus contradiction). But I want to prove now that the only knots with crossing number $3$ are the two trefoils knots. Surely both are inside the set of knots with crossing number 3, but the other site?! Or is this just trying?? Look up also to a torus link $T_{p,q}$. I want to prove this statement: $T_{p,q}$ is a knot (thus not a link) if and only if $p$ and $q$ are coprime. Can someone help me?

Answer 1

Hint 1: A knot diagram is a four-valent planar graph with crossing information for each vertex, so consider valence-four planar graphs. What sort of crossing information gives a knot? What knots can arise this way?

Hint 2: Recall that the definition of a torus knot is the image of a closed curve on the surface under a smooth embedding of the torus into $\mathbb{R}^3$. If we regard the torus as $\mathbb{R}^2/\mathbb{Z}^2$, all closed curves are homotopic to the image of a straight lines under identification. The coefficients $p,q$ are related to the slope of the line. Can you relate $p$ and $q$ to the number of components of the image of the line?