We say a knot in $S^3$ has tunnel number $1$ if there is an arc $\alpha$ in $S^3 - K$ with both endpoints on $K$, such that $S^3 - N(K \cup \alpha)$ is a genus $2$ handlebody (here $N$ denotes a small thickened neighborhood of the graph $K \cup \alpha$). I have two questions.
- If $K$ has tunnel number $1$, is there a presentation for $\pi_1(S^3 - K)$ with two generators and one relation?
- What is the tunnel number of the figure eight knot?
Here is a post on MathOverflow which answers your first question, by Ian Agol.
As to the second question, we know that the figure-eight knot is a 2-bridge knot. Using the fact that $t(K)\leq b(K)-1$, we immediately have it is a tunnel number one knot. This relation comes up in this paper by Morimoto, Sakuma, and Yokota.
The quick way to see the figure-eight knot is a tunnel number one knot is to draw its standard projection and place your arc (Blue) at the top between the two maxima. Then untwist the two crossings in the middle (Oops, I drew the untwisting arrow in the wrong direction. The red arrow should go to the right, not the left). And then slide the other two off. Hope this helps.