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I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let $D$ be the disc and $T=S^{1}\times D$ the solid torus. Then I would like to consider smooth immersions $f:T\to S^{3}$ with the property that:

  1. $f\mid_{S^{1}}:S^{1}\to S^{3}$ (here $S^{1}$ is the first factor in $T$) is a smooth immersion.
  2. $f\mid_{S^{1}}$ has only finitely many double points.
  3. The crossings of $f\mid_{S^{1}}$ are transverse.
  4. $f\mid_{S^{1}}$ has no $k$-points for $k>2$.

One can make sense of ambient isotopy in the usual way. Such maps have been studied by various authors in the absense of a framing, i.e. just examining $f\mid_{S^{1}}$. For instance:

Joan S. Birman, "New Points of View in Knot Theory," Bulletin of the American Mathematical Society, Volume 28, Number 2, April 1993

There the Reidemeister moves for singular knots are given, and they are exactly what you would expect. The proof, of course, relies on the usual singularity theory of Arnol'd. My question is:

Question: Is anyone aware of a similar result for framed singular knots?

In principle I can return to Arnol'd's singularity theory and attempt to rederive a Reidemeister theorem, but I'd like to avoid re-inventing the wheel if possible.

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  • $\begingroup$ The framings you are considering are not the standard framings on singular knots. Typically, you would pin down the framings at the singularities, looking at something more like an embedding of a ribbon with 4-valent vertices. In your framings, a "twist" is allowed to pass through a singularity, so there is only $\mathbb{Z}$ worth of framings for a given unframed singular knot. $\endgroup$
    – Dylan Thurston
    Commented Jun 17, 2015 at 10:53
  • $\begingroup$ I don't think this is standard. The Reidemeister moves will be easy to work out, of course. You may find it easier to work with PL knots (as Reidemeister did), rather than working through the singularity theory. $\endgroup$
    – Dylan Thurston
    Commented Jun 17, 2015 at 10:54
  • $\begingroup$ @Dylan hmmm I certainly don't want the framing to pass through the singularities in S^3. Could you say a bit more about what the standard framing of a singular knot is, or provide a reference? $\endgroup$
    – Paul
    Commented Jun 17, 2015 at 14:54
  • $\begingroup$ @Dylan maybe something like this corrects for your comment: The image of $S^{1}\times\{0\}\times\{1\}$ provides a framing. So we further require that at each double point of $f$, the angle between the frames of the two strands to be $\pi$. $\endgroup$
    – Paul
    Commented Jun 17, 2015 at 15:15
  • $\begingroup$ I don't quite know what you mean by "the angle between the frames", but something like that should be what I meant. $\endgroup$
    – Dylan Thurston
    Commented Jun 18, 2015 at 1:07

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