The number of framings of a knot in a 3-manifold

Abstract

In view of the self-linking invariant, the number |K| of framed knots in S3 with given underlying knot K is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that |K| is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of S1 × S2; the knot K need not be zero-homologous and the manifold is not required to be compact. We show that when M is orientable, the number |K| is infinite unless K intersects a nonseparating sphere at exactly one point, in which case |K| = 2; the existence of a nonseparating sphere implies that M contains a connected sum factor of S1 × S2. For knots in nonorientable manifolds we show that if |K| is finite, then K is disorienting, or there is an orientation-preserving isotopy of the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded S2 or ℝP2 at exactly one point, or it intersects some embedded S2 at exactly two points in such a way that a closed curve consisting of an arc in K between the intersection points and an arc in S2 is disorienting.