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

TL;DR

This paper investigates the conditions under which the number of framings of a knot in a 3-manifold is finite or infinite, revealing topological constraints related to sphere intersections and manifold orientability.

Contribution

It characterizes when the set of framings of a knot in a 3-manifold is finite or infinite, extending previous invariants to non-compact and non-zero-homologous cases.

Findings

Number of framings is infinite unless the knot intersects a non-separating sphere at exactly one point.

In orientable manifolds, the finiteness of framings relates to intersection with non-separating spheres.

In nonorientable manifolds, finiteness implies specific disorienting properties or intersections with spheres or projective planes.

Abstract

In view of the self-linking invariant, the number $|K|$ of framed knots in $S^3$ 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 $S^1\times S^2$; 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 non-separating sphere at exactly one point, in which case $|K|=2$; the existence of a non-separating sphere implies that $M$ contains a connected sum factor of $S^1\times S^2$. For knots in nonorientable manifolds we show that if $|K|$ is finite, then $K$ is disorienting, or there is an isotopy from the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded $S^2$ or $\mathbb R P^2$ at exactly one point, or it intersects some embedded $S^2$ 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 $S^2$ is disorienting.