Lattice-polygon knot-statistics asymptotic conjecture

For a knot type $K$, let $p_n(K)$ be the number of $n$-edge polygons of knot type $K$. Let $0_1$ be the unknot, let $f_K$ be the number of prime knot factors in the knot decomposition of $K$, and set $f_{0_1}=0$. Let $\alpha_0$ be the unknot entropic critical exponent and let $\mu_{0_1}$ be the unknot exponential growth constant. Lattice-polygon knot-statistics conjecture. As $n\to\infty$, there are constants $B_K$, $A_0$, and $\mu_{0_1}$, potentially dependent on the lattice, such that

$$p_n(K)=B_K n^{f_K}p_n(0_1)(1+o(1)),$$

and

$$p_n(0_1)=A_0n^{\alpha_0}\mu_{0_1}^n(1+o(1)).$$

The conjecture follows from polymer scaling theory and numerical evidence. Its form is proved in the tube setting discussed later when the relevant tube asymptotics are available, but it is not established in general lattice-polygon models.