Summary: A knot \(k\) in \(S^3\) has property \(Q\) if there is a closed surface \(S\) in \(S^3\) containing \(k\) such that \(k\) is imprimitive in \(H_1A\) and \(H_1B\) where \(A\) and \(B\) are the closures of the components of \(S^3-S\). If, in adition, \(lk(k,k^+)\neq 0\), where \(k^+\) is contained in \(S\) and is parallel to \(k\), then \(k\) has property \(Q^*\). We determine the 2-bridge knots \(K({\beta\over \alpha})\) that have property \(Q\) in terms of a continued fraction expansion of \({\alpha\over \beta}\). We also prove that the following conditions are equivalent: (i) \(K({\beta \over\alpha})\) has property \(Q\); (ii) \(K({\beta\over\alpha})\) has property \(Q^*\); (iii) There is an epimorphism from the group of \(K({\beta \over\alpha})\) onto a free product; (iv) There is an epimorphism from the group of \(K({\beta \over\alpha})\) onto \(\mathbb{Z}_2* \mathbb{Z}_d\).