Abstract

Given a three-manifold M and a cohomology class τ in H¹(M,Z ∕ nZ), there is a naturally defined invariant of singular knots in M with exactly one double point, V _τ. It has been known that for some manifolds V _τ is integrable and that in these cases it defines an easily computed and highly effective knot invariant. This paper provides necessary and suficient conditions on M for the integrability of V _τ. The class of manifolds for which V _τ is integrable (regardless of the choice of τ) is shown to include all hyperbolic manifolds, all complements of knots in irreducible homology spheres, all irreducible Z ∕ 2Z-homology spheres, and most Seifert-fibered manifolds.

Authors