Epimorphisms of knot groups onto free products
Imposing conditions on a closed surface in \(S^3\) containing a knot \(k\) leads to the definition of properties Q, Q*, and Q**. Roughly speaking, property Q is a crude generalization of torus knots. The following theorem is proved and is representative of other results in the paper. The knot \(k\) has property Q iff any one of the following is true: (1) There exists an epimorphism from the group of the knot to \(\mathbb{Z}_a*\mathbb{Z}_b\) sending a meridian to an element of length 2. (2) There exists an \(H_1\) splitting surface for \(k\) whose boundary has slope with absolute value 2. (3) There exists an incompressible \(H_1\) splitting surface for \(k\) whose boundary has slope with absolute value 2. (4) There exists an epimorphism from the group of the knot to the group of a torus knot sending a meridian to a meridian. A corollary to this theorem is that if a knot has property Q then its Alexander Polynomial is divisible by the Alexander Polynomial of a torus knot. The paper contains a number of examples.
scientific article; zbMATH DE number 1956002