I know that in general, it is difficult to tell whether a knot is prime or not. However, the Wikipedia page has established that the trefoil knot and the figure 8 knot are prime knots.
I've managed to draw the Seifert circles for these two knots and obtained the genus of their Seifert surfaces.
For the trefoil knot, I got $g(S_1)=(3-2+1)/2=1$ and for the figure 8 knot, I got $g(S_2)=(4-2+1)/2=3/2$. I know I'm supposed to use Seifert's Theorem that $g(K\#L)=g(K)+g(L)$ somehow, but what can I do?
You should find that the figure 8 knot (with its standard knot diagram) has 4 crossings, 3 Seifert circles and 1 component which gives a knot genus of $(2+4-3-1)/2=1$. Given that knot genus is a natural number, and the trefoil and figure 8 knots are not trivial, we can deduce that they both have genus 1. If they were not prime, then there would exist two knots of positive integer genus (as the unknot is not prime), whose geni add to give 1, by Seifert's Theorem. This is clearly a contradiction.
