# Positive genus and Abelian-ness

So if a Seifert surface of a knot K has positive genus, what can we say about the its complement's first fundamental group, is it Abelian because the only knot with Seifert surface of positive knot is the unknot?

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The only knot whose complement has an abelian fundamental group is the unknot. The fundamental group of a knot complement abelianizes to $\mathbb Z$ by Alexander Duality. So in fact the only possible abelian fundamental group for a knot complement is $\mathbb Z$. It is known that the unknot is the only knot where $\pi_1$ of the complement is $\mathbb Z$.