Genus-zero links with prescribed knots as components

Abstract

We prove that any finite collection of at least three isotopy classes of knots in a 3-manifold $M$ is realizable as the components of a genus-zero link in $M$, provided that an obvious requirement on their conjugacy classes in ${\pi }_1(M)$ is met. This condition is vacuously satisfied for $M={\mathbb{S}}^3$, and in this case we also control the pairwise linking numbers of the components. Replacing the 3-genus with the 4-genus, we obtain an analogous result where only two knot isotopy classes are prescribed.

arXiv comment15 pages, 11 figures, comments welcome