Genus-zero links with prescribed knots as components

Raphael Appenzeller; JosÃ© Pedro Quintanilha

geometric topology

Genus-zero links with prescribed knots as components

arXiv:2607.12729

summary

The authors prove that any finite collection of at least three knot isotopy classes in a 3‑manifold can be realized as the components of a genus‑zero link, provided a natural condition on their conjugacy classes in the fundamental group holds; in S³ the construction also controls pairwise linking numbers, and an analogous result for two knots holds when using the 4‑genus.

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 $I ¨ _{1} (M)$ is met. This condition is vacuously satisfied for $M = S^{3}$ , 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.

15 pages, 11 figures, comments welcome

Topics & keywords

#knot theory #link theory #3-manifolds #genus-zero links #linking numbersgenus-zero linkisotopy classconjugacy classπ1(M)4-genus