Let $K\subseteq S^3$ be a knot.
In real-life, knots (like protein chains) $K$ moves around stochastically, and experimentally the lowest energy/highest entropy states are particularly simple from a knot-theoretic point of view (see e.g. [BOS] or this youtube video).
Question 1: What are some natural candidates(s) for an energy function $E(K)$, explicitly?
The following is more of a physics question:
Question 2: Does the partition function $\mathcal{Z}=\int_K e^{E(K)/T}$ define 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, and apply it to the case where you want to model the position of a single knot. A naive guess is that this might have to do with "line operators" in some SFT.