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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?

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up vote 3 down vote accepted

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.

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How do you get 3 Seifert circles for the figure 8 knot? I used the diagram from Wikipedia (which I hope is the standard diagram) and only managed to get 2 - the topmost "hole" and the bottom-middle "hole". Apologies if that wasn't the best descrption. Also, isn't the formula $g(S)=(c-s+1)/2? Where did your number 2 come in? – Haikal Yeo May 3 '13 at 18:29
I used Wikipedia's definition of genus=(2 + #crossings - #seifert_circles - #knot_components)/2 If $K$ is a knot, this becomes genus=(1 + #crossings - #seifert_circles)/2. Without drawing a picture I can't really show how I got 3 circles but I guess I got one small circle at the top, and then one big circle with another smaller circle on the inside. That's the best way I can describe it. I can draw a picture if it's still not clear but I would suggest drawing it again for yourself and carefully going through the crossing switches. – Dan Rust May 3 '13 at 18:43

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