Singular framing of a singular knot

Question

In "Introduction to Vassiliev Knot Invariants" (https://arxiv.org/abs/1103.5628), singular framing is defined in p.82, and I think that we can twist the framing arbitrarily around zero points of the framing in this definition, so I think we have to impose some extra conditions on the definition to avoid such deformation. Is it correct? And if so, are there any extra conditions widely used?

Answer 1

Yes, that is correct that you must impose some extra conditions. They need to be "simple" singularities, in the same sort of sense that the usual singular points for Vassiliev invariants are "simple" singularities.

The original idea for Vassiliev was to study the space of smooth embeddings $\operatorname{Emb}(S^1,\mathbb{R}^3)$, which is the complement of the "discriminant" $\Sigma\subset C^\infty(S^1,\mathbb{R}^3)$ of the smooth non-embeddings. In some sense $\Sigma$ is a codimension-$1$ subspace of $C^\infty(S^1,\mathbb{R}^3)$. Inside $\Sigma$ is $\Sigma'$, the smooth maps that are immersions such that at their single point of self-intersection they intersect transversely. This is a codimension-$1$ subspace of $C^\infty(S^1,\mathbb{R}^3)$, and $\Sigma\setminus\Sigma'$ is a codimension-$2$ subspace. Thinking of a smooth homotopy as a path through $C^\infty(S^1,\mathbb{R}^3)$ that can be regarded as an element of from $C^\infty(S^1\times[0,1],\mathbb{R}^3)$, the set of those homotopies that intersect $\Sigma$ transversely is a dense subset. Furthermore, since the codimension is $1$ it does so at isolated points in $[0,1]$ and it only ever intersects $\Sigma'$. This is why we only need to consider transverse self-intersections.

Let's now consider what a framed knot is. One definition is an embedding $f:S^1\times \mathbb{R}^3$ along with a nonvanishing section of the normal bundle $s:S^1\to T\mathbb{R}^3/f_*TS^1$. We can still make sense of this for $f\in \operatorname{Emb}(S^1,\mathbb{R}^3)\cup \Sigma'$, since maps in $\Sigma'$ are still immersions. (Note: to do this "right", I think it would be better to consider localizing embeddings of solid tori $S^1\times D^2$ at their core $S^1\times\{0\}$; that is, by taking maps in $\operatorname{Emb}(S^1\times D^2,\mathbb{R}^3)$ modulo restricting to open subsets of $S^1\times\{0\}$. Then we can consider the possible singularities of these in a uniform way, rather than adding framing to a previous analysis.) We can consider the space of all sections, and it turns out ones that vanish somewhere form a codimension-$1$ subspace again. The sections that intersect the zero section non-transversely or more than once form a codimension-$2$ space, so we don't need to consider them. That leaves the sections that intersect the zero section transversely exactly once. Here's an illustration of the images of sections in a small neighborhood of a knot:

Sections intersecting the zero section transversely have exactly two "resolutions":

Topologically speaking, that means they are in the closures of exactly two connected components of the space of all framed knots. Sections intersecting non-transversely have "too many" resolutions --- they're in the closure of each connected component for each framing of a given knot.

So, the reason why you can't twist the framing arbitrarily around a point where the framing vanishes is that, to do so in a smooth way, you would need to have it pass through moments where the framing is not transverse to the zero section.

---

Another way to model all this is that a framed knot is a knot along with a very-close-by knot isotopic to it in a tubular neighborhood, like the relationship between the black and blue curves in the illustrations above. Then you consider Vassiliev-like singularities of two types: transverse self-intersections of the black curves, and transverse intersections of the black curve and the blue curve between "corresponding" points.

---

It might also be worth mentioning some motivation for all of this, to see why considering only these simple singularities is what you want to do. If you want to study framed knots up to isotopy, what you can do is consider the space $F$ of all framed knots and try to compute the connected components. One way to do this would be consider the set of all a locally-constant functions $v:F\to R$ for some ring $R$, also known as $H^0(F;R)$. Vassiliev's idea, very roughly, is to try to realize these functions as being the "integral" of a "vector field" on the space of singular framed knots (ones that allow singularities in both the embedding and the framing). Given the value of $v$ on the unknot, you can recover its value for other knots by taking a "path integral" from the unknot to a given knot in this extended space, but by the above analysis we can assume the path goes through knots with only simple singularities at isolated times. It is at these points that the value of the integral jumps in value, which is the Vassiliev relation (which relies on the fact that, since the knots are oriented and crossings have a sign, we are able to assign an orientation to the jump).