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 $f\mid_{S^{1}}:S^{1}\to S^{3}$ (here $S^{1}$ is the first factor in $T$) is a smooth immersion such that:

1. $f$ has only finitely many double points.
2. The crossings are transverse.
3. $f$ 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.

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.