I am wondering if the mean curvature flow of one-dimensional submanifolds of $\mathbb{R}^3$ is understood well enough to give some perspective on (and hopefully a proof of) something like the Fary-Milnor theorem.

For reference, the Fenchel theorem (1929) says that if $c:S^1\to\mathbb{R}^3$ is a smooth embedding, then the total curvature is at least $2\pi$. The Fary-Milnor theorem (1949/50) says that if $c$ forms a nontrivial knot, then the total curvature is at least $4\pi$.

Steven Altschuler ("Singularities of the curve shrinking flow for space curves", JDG 1991) showed that if $c_t:S^1\to\mathbb{R}^3$ is a one-parameter family of smooth embeddings satisfying the mean curvature flow, then $$\frac{d}{dt}\int_{S^1}|\kappa_t|\leq -\int_{S^1}\tau_t^2|\kappa_t|$$ where $\tau_t$ is the torsion of $c_t$. So if $\int_{S^1}|\kappa_0|<4\pi$ then certainly $\int_{S^1}|\kappa_t|<4\pi$ for all positive $t$. So it seems like one could hope for a proof of the Fary-Milnor theorem which is more or less directly analogous to Hamilton-Perelman's proof of the Poincaré conjecture or of the topological classification of closed 3-manifolds with nonnegative and positive scalar curvature.

The problem would rest on understanding the singularities of the mean curvature flow. Altschuler seems to have shown that a singularity of the mean curvature flow in this setting is characterized by blowup of the curvature (just like Ricci flow), and that a nontrivial tangent flow is given either by an Abresch-Langer solution or the grim reaper solution. This seems to be directly parallel to the Perelman-Brendle theorem which says that the analogous blowup limit of a finite-time singularity of a Ricci flow on a compact 3-manifold is either a quotient of shrinking round cylinders or the Bryant soliton, which was (in a weaker version) Perelman's breakthrough result.

So it seems like the key ingredients are there. Can they be put together? It seems like the basic problem is that I don't know what the "surgery" analogue might look like or how it might be relevant.

So, more generally, forgetting about Fary-Milnor theorem in particular, could one hope to use mean curvature flow for any sort of application in knot theory? Perhaps the proper analogue of the Hamilton-Perelman approach would decompose a given knot as a connected sum of prime knots, and would give some canonical representation thereof? This seems to be comparable to the geometrization conjecture, although the existence and uniqueness of such a decomposition seems to be already known in knot theory.

In order that I have a reasonably concrete question:

- Ricci flow with surgery on 3D compact manifolds is closely analogous to mean curvature flow of mean-convex surfaces in $\mathbb{R}^3$ (Brendle-Huisken). Is there an analogy, or a conjectural analogy, to mean curvature flow of curves in $\mathbb{R}^3$, or in 3-manifolds? Are there (conjectural) applications in knot theory?

I couldn't find any references on the web. I am aware of Perelman's use of mean curvature of curves in a Ricci flow background to show finite extinction time of Ricci flow on simply-connected 3-manifolds, but the only detailed version which I know to exist of this, Morgan-Tian's book, seems to have some basic errors (cf. Bahri "Five gaps in mathematics" and some followups on the arxiv)