What is the energy of a knot?

Physicists, mathematicians, people who study protein folding, etc., are all in theory interested in knots moving in $$\mathbf{R}^3$$:

To try to understand them physically, the first question is:

Question 1: What are the natural notion(s) of energy $$E(K)$$ to assign to a knot?

In the resulting physics, you would like moving towards knots with the lowest energy/highest entropy to converge to particularly "simple" knots, e.g. [BOS] or this youtube video. For example, you would see that the above is actually the trivial knot!

In the literature, I only find cryptic statements like in [G]:

The topology-oriented reader may be wondering as to what is the energy $$E(K)$$. Physically, it can be either attractive or repulsive interaction between parts of the polymer that come close to each other in space, although they may be arbitrarily far apart along the chain contour with $$0 < s < 1$$ fraction of total length. distances. I shall not explain more about the energy; I mention that physicists believe to have a pretty good understanding of that part, and this is why I shall concentrate on the (mathematically) simplest case when $$E(K) = 0$$.

but no even heuristics of what form you might expect $$E(K)$$ to take. Having answered this it's natural to wonder

Question 2: What's the relevant physics here? Is this a Statistical Field Theory? If so, in what dimension- three ($$=\dim \mathbf{R}^3$$), two ($$=\dim \mathbf{R}^3-\dim K$$) or infinite ($$=$$ dimension of space of possible $$K\subseteq\mathbf{R}^3$$'s)? Why?

In particular, if so it seems confusing to me that you can take SFT, designed to model statistical properties of large numbers of point particles/objects/strings(?), and apply it to the case where you want to model the position of a single knot.

[G] Grosberg, A.Y., 1998. Entropy of a knot: simple arguments about difficult problem. In Ideal Knots (pp. 129-142).

[BOS] Baiesi, M., Orlandini, E. and Stella, A.L., 2010. The entropic cost to tie a knot. Journal of Statistical Mechanics: Theory and Experiment, 2010(06), p.P06012.

• This is probably more suited for math.stackexchange; you can find some links here though (en.wikipedia.org/wiki/Knot_energy) Jul 24, 2023 at 22:17
• This isn't for protein folding, but it provides a quantitative picture of how the energies of physical knots can the calculated: sciencedirect.com/science/article/pii/…
– Buzz
Jul 25, 2023 at 20:38

• I think you misunderstood the thrust of my question, Q1 is looking for explicit expressions for $E(K)$ and Q2 for a discussion of the SFT. Jul 24, 2023 at 22:25