The entropic cost to tie a knot

We estimate by Monte Carlo simulations the configurational entropy of $N$-steps polygons in the cubic lattice with fixed knot type. By collecting a rich statistics of configurations with very large values of $N$ we are able to analyse the asymptotic behaviour of the partition function of the problem for different knot types. Our results confirm that, in the large $N$ limit, each prime knot is localized in a small region of the polygon, regardless of the possible presence of other knots. Each prime knot component may slide along the unknotted region contributing to the overall configurational entropy with a term proportional to $\ln N$. Furthermore, we discover that the mere existence of a knot requires a well defined entropic cost that scales exponentially with its minimal length. In the case of polygons with composite knots it turns out that the partition function can be simply factorized in terms that depend only on prime components with an additional combinatorial factor that takes into account the statistical property that by interchanging two identical prime knot components in the polygon the corresponding set of overall configuration remains unaltered. Finally, the above results allow to conjecture a sequence of inequalities for the connective constants of polygons whose topology varies within a given family of composite knot types.

💡 Research Summary

The paper investigates the configurational entropy of self‑avoiding polygons (SAPs) on the cubic lattice when the knot type is fixed. Using a combination of the BFACF algorithm and a two‑pivot Monte Carlo scheme, the authors generate unbiased samples of SAPs with up to 200 000 steps, far beyond previous studies. After each polygon is generated, a sophisticated reduction procedure—iterative length‑reduction preserving topology, selection of the planar projection with minimal crossings, and Dowker‑code analysis with Reidemeister‑type moves—is applied to identify the knot type, even for highly tangled configurations. This enables the collection of extensive statistics for both prime knots (up to six crossings) and composite knots with up to five prime components.

The authors first revisit the well‑known scaling law for unrestricted SAPs, Z(N)≈A μ^N N^{α−2}, and for unknotted polygons, Z_∅(N)≈A_∅ μ_∅^N N^{α_∅−2}. By fitting the decay of the unknotting probability P_∅(N)=Z_∅(N)/Z(N) they find α=α_∅ (within statistical error) and A≈A_∅, while the difference between the connective constants μ and μ_∅ is extremely small (μ−μ_∅≈2×10⁻⁵), corresponding to a characteristic length N₀≈2.1×10⁵.

The central contribution is the proposal and verification of a refined asymptotic form for the partition function of polygons with a fixed knot type k:

 Z_k(N) ≈ A_∅ C_k μ_∅^{N} N^{α_∅−2+π_k},

where π_k is the number of prime components in the knot decomposition and C_k is an “entropic cost” associated with tying the knot. For a prime knot (π_k=1) the ratio Z_k(N)/Z_∅(N) scales as N/C_k, confirming that each prime knot contributes a factor N (or equivalently a term ln N in the entropy) due to its ability to slide along the unknotted backbone.

A striking empirical finding is that C_k grows exponentially with the minimal lattice length ℓ_k required to realize the knot on the cubic lattice:

 C_k ≈ μ^{ℓ_k/3}.

Thus the entropic penalty can be interpreted as a loss of V_k = ℓ_k/3 monomers that must be “used” to form the knot. For example, the trefoil (3₁) has ℓ=24, giving V=8 and C≈μ⁸≈10⁶; the 6₁ knot has ℓ=40, giving V≈13.3 and C≈μ^{13.3}≈10⁸. This relation holds for all prime knots examined (3₁, 4₁, 5₁, 5₂, 6₁, 6₂, 6₃) and suggests a universal link between a knot’s minimal lattice representation and its statistical weight in a swollen polymer ring.

The analysis is extended to composite knots. Assuming independence of prime components, the authors derive a factorized expression:

 Z_{k₁#k₂…#k_{π}}(N) ≈ N^{π} (∏{i=1}^{π} C{k_i}) μ_∅^{N} N^{α_∅−2},

with an additional combinatorial factor 1/(m_i!) for each set of m_i identical prime components, accounting for the indistinguishability of swapping identical knots along the chain. This predicts, for instance, that a polygon containing two trefoils (3₁#3₁) has Z≈(N²/2) C_{3₁}² μ_∅^{N} N^{α_∅−2}. The numerical data for several composite knots (e.g., 3₁#5₁, 3₁#3₁#4₁) confirm the N^{π} scaling and the product form of the entropic costs.

Finally, the paper proposes a hierarchy of connective constants for different knot families. Because the exponential factor μ_∅^{N} is common, the differences arise solely from the power of N (π_k) and the prefactors C_k. The authors conjecture that for any two knot types k and k′ with π_k ≤ π_{k′} and C_k ≤ C_{k′}, the corresponding connective constants satisfy μ_k ≤ μ_{k′}. This yields a set of inequalities linking the growth rates of polygon counts across families of composite knots.

In summary, the work provides a comprehensive quantitative framework for the entropy loss associated with knotting in lattice polymers. It demonstrates that (i) each prime knot behaves as a localized, sliding defect contributing a ln N term; (ii) the cost of creating a specific knot is exponentially tied to its minimal lattice length; (iii) composite knots factorize into independent prime contributions with simple combinatorial corrections; and (iv) these results lead to testable predictions about the relative abundances of different knot types in long polymer rings. The methodology and scaling relations open avenues for studying knot statistics in other lattice geometries, off‑lattice models, and experimentally relevant systems such as circular DNA under various solvent conditions.