Torus knots are prime

For coprime p,q with absolute values at least two, the knot group is $⟨a,b\mid a^p=b^q⟩$. Its nontrivial central element $z=a^p=b^q$ maps nontrivially to the abelianization. A meridian $a^u{b}^v$, where $qu+pv=1$, has infinite order in the quotient $C_{|p|}\ast C_{|q|}$. A composite knot group would be an amalgamated free product of proper factors over its meridian subgroup, whose center lies in that subgroup. Then z would be a meridian power, contradicting the infinite order of the meridian modulo z.

// Target: geometric-group-theory.bigb