= Torus knots are prime
For coprime p,q with absolute values at least two, the <knot group> is $\langle a,b\mid a^p=b^q\rangle$. Its nontrivial central element $z=a^p=b^q$ maps nontrivially to the abelianization. A meridian $a^ub^v$, where $qu+pv=1$, has infinite order in the quotient $C_{|p|}*C_{|q|}$. 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
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