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

enter image description here

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.


1 Answer 1


If we are to consider the knot a real polymer, then it has constituents such as amino acids, nucleotides, etc.(I don't know much biology). Therefore, the total energy of the knot is a sum of the energy of the constituents. This is a very tedious thing and would be more likely determined experimentally or approximately by taking an integral of a theoretical Hamiltonian density.

If we're to consider the knot an abstract object with no constituents, similar to a string in string theory, then we would find the energy of the knot like we find the energy of any other object, its kinetic energy (dependent on momentum) plus its potential energy (dependent on position).

Without more information about a specific knot, it's hard to give a more detailed answer. If we are to consider this a string as in string theory, then it doesn't make a lot of sense to ask the energy of it, as the strings more so give way to fields which may assign energies to the excitations created by the strings, but not of an individual string. In the case of the constituents, the only way to give a more detailed answer which is still as general as the knot you're asking about would be to essentially copy-paste the standard model Lagrangian or detail the field equations for each constituent and their interactions, but those wouldn't do you any good and aren't at all specific to the knot but more so matter in general. It would also be horribly taxing.

  • $\begingroup$ 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. $\endgroup$
    – Pulcinella
    Jul 24, 2023 at 22:25
  • $\begingroup$ @Pulcinella Q1: E(K)=T(K)+V(K), where T is the kinetic function and V is the potential function. If the knot has constituents and we're to consider it classically, take E(K)=int(E(s))ds from s=0 to s=s, where E(s)=T(s)+V(s). Quantizing this simply eliminates the values of s for which an excitation is not present. Q2: Classical mechanics 3D in Euclidean space, or QM infinite dim in Hilbert space with QFT 4D in Minkowskian spacetime. What that all means practically for the problem is more or less what my answer consists of. $\endgroup$ Jul 24, 2023 at 22:58
  • $\begingroup$ To pull out two things from this answer to help explain why Q1 asking for an explicit formula is underdetermined... (a) the kinetic energy will generically depend on the state of motion of the knot, so the energy depends not just on the knot is embedded in space but on the (partial) time derivative describing how the knot moves in space, (b) the "potential energy" component that does depend only on the knot's shape will depend on the specific application you have in mind (such as polymers vs fundamental relativistic strings). $\endgroup$
    – Andrew
    Jul 24, 2023 at 23:38
  • $\begingroup$ @Andrew The dynamics of the knot can be used to find its energy, but the starting point for this is usually the Lagrangian which already requires an understanding of its energy, so it feels circular. Without the Lagrangian or experimentation, I'm not sure how to accomplish what you're saying, which is why I feel what I said is the closest you can get to an answer. On the potential energy, I agree and took that as implied when answering. I suppose explicitly stating it would make sense given the nature of the question however, so thank you. $\endgroup$ Jul 24, 2023 at 23:52
  • $\begingroup$ @Joseph_Kopp Yep I totally agree with you, the intent of my comment was to point out to the OP that getting an explicit formula beyond what you give is complicated. $\endgroup$
    – Andrew
    Jul 25, 2023 at 0:01

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