Entanglement statistics of polymers in a lattice tube and unknotting of 4-plats

TL;DR

The paper proves the Knot Entropy Conjecture for lattice polygons confined to the smallest nontrivial lattice tube _T^*, establishing that for any fixed non-split link L embeddable in _T^*, p_{_T^*,n}(L) grows exponentially with the same rate as unknot polygons p_{_T^*,n}(0_1) and with a power-law correction determined by the number of prime factors of L. The main strategy combines a diagrammatic upper-bound approach via unknotting 4-plat diagrams (using 3- and 4-braid insertions) with a robust lower-bound pattern theorem for unknots derived from exact transfer-matrix calculations, yielding precise bounds that differ only by constant factors. This yields a full KE-type asymptotic for fixed-link-type embeddings in the tube and a general pattern theorem for fixed-link-type embeddings, implying knotting/linking is typically localized in confinement. The results connect knot theory with polymer physics, offering rigorous insight into DNA in nanochannels and broader implications for knot localization and entropic statistics in confined polymers.

Abstract

The Knot Entropy Conjecture states that the exponential growth rate of the number of $n$-edge lattice polygons with knot-type $K$ is the same as that for unknot polygons. Moreover, the next order growth follows a power law in $n$ with an exponent that increases by one for each prime knot in the knot decomposition of $K$. We provide the first proof of this conjecture by considering knots and non-split links in tube $\mathbb{T}^*$, an $\infty \times 2\times 1$ sublattice of the simple cubic lattice. We establish upper and lower bounds relating the asymptotics of the number of $n$-edge polygons with fixed link-type in $\mathbb{T}^*$ to that of the number of $n$-edge unknots. For the upper bound, we prove that polygons can be unknotted by braid insertions. For the lower bound, we prove a pattern theorem for unknots using information from exact transfer-matrices. This work provides new knot theory results for 4-plats and new combinatorics results for lattice polygons. Connections to modelling polymers such as DNA in nanochannels are highlighted.

Figures (19)

• Figure 1: A.Example for the upper bound in Theorem \ref{['thm-bounds']}, equation \ref{['mainrelation']}. A 98-edge polygon of knot-type $3_1$ (trefoil) in $\mathbb T^*$; B. An insert enclosed in a 3-block. The insert is a 4-braid containing a half twist of two strings. When inserted into (A) at the identified location, this 3-block converts the $3_1$ into an unknot. C.Example for the lower bound in Theorem \ref{['thm-bounds']}, equation \ref{['mainrelation']}. The figure shows an unknot lattice polygon with eight highlighted 2-sections. D. The shaded image is a trefoil pattern enclosed in a $7$-block. When this pattern is inserted at any of the 2-sections indicated by arrows, the unknotted polygon turns into a trefoil polygon in $\mathbb T^*$. Note that one can insert the trefoil pattern at any of the other 2-sections of the original polygon by modifying its ends. Figures (A) to (D) have been modified from beaton_first_2024.
• Figure 2: A. A 98-edge polygon of knot-type $3_1$ (trefoil) in $\mathbb T^*$; B. A shifted knot diagram obtained from a 2-dimensional projection of the polygon in (A); and C. The 4-plat diagram corresponding to the knot in (A) and the knot diagram in (B). D. A 4-plat diagram of the unknot obtained by inserting a half twist at the location indicated with a dotted line in (B) and in (C). Figures (A) to (D) have been modified from Fig. 1 in beaton_first_2024. Figures (A) to (D) have been modified from beaton_first_2024.
• Figure 3: The diagrams in this figure illustrate the elementary $4$-braids $\sigma_1,\sigma_2,\sigma_3, \sigma_1^{-1},\sigma_2^{-1}$ and $\sigma_3^{-1}$ (middle), and the closures $[_1$, $[_2$, $]_1$, $]_2$.
• Figure 4: Insertions of $w_0=\sigma_2^{-2}$ and $w_0=\sigma_2^{-1}\sigma_1^{-1}$ on $4$-plat diagrams. These insertions are used to obtain diagrams of the unknot.
• Figure 5: Illustrations of the $\mathcal{A}$ and $\mathcal{B}$ moves which take one 4-plat diagram to another. The $A$ in the $\mathcal{A}_3$ move and the $B$ in the $\mathcal{B}_3$ move are 3-braid diagrams. The $A$ in the $\mathcal{A}_2$ move is a 4-braid diagram. Every other rectangular block represents a 4-braid diagram along with an arbitrary one-sided closure. An $\mathcal{A}_1$-move relates two diagrams of the form $[_iw_1\sigma_{1}^{\epsilon}w_2]_j$ and $[_iw_1\sigma_{3}^{\epsilon}w_2]_j$ for $\epsilon=\pm1$. An $\mathcal{A}_2$-move relates two diagrams of the form $[_iw]_j$ and $[_j\overline{w}]_i$, where $\overline{w}$ is the reverse word of $w$ as defined above. An $\mathcal{A}_3$-move relates two diagrams of the form $[_1w]_1$ and $[_2\widehat{w}]_2$, or $[_1w]_2$ and $[_2\widehat{w}]_1$ for a $3$-braid word $w$, where $\widehat{w}$ is the $3$-braid word which is the flipped word defined above. An $\mathcal{A}_4$-move relates two diagrams of the form $[_iw\sigma_1^{\epsilon}]_1$ and $[_iw\sigma_2^{-\epsilon}]_2$ for $\epsilon=\pm1$. A $\mathcal{B}_1$ move is a Reidemeister I move on one end of a $4$-plat diagram, that deforms $[_i\sigma_k^{\epsilon}w]_j$ into $[_iw]_j$ or deforms $[_jw\sigma_k^{\epsilon}]_i$ into $[_jw]_i$ for $\epsilon=\pm1$ and $(k,i)=(1,2),(2,1)$ or $(3,2)$. A $\mathcal{B}_2$ move is a Reidemeister II move on a reducible $4$-braid diagram, that deforms $[_iw_1\sigma_k^{\epsilon}\sigma_k^{-\epsilon}w_2]_j$ into $[_iw_1w_2]_j$ for $\epsilon=\pm1$ and $i,j,k\in\{1,2,3\}$. A $\mathcal{B}_3$ move deforms $[_iw_1\sigma_1^{\epsilon}\sigma_2^{\epsilon} w_2]_j$ into $[_iw_1\sigma_2^{-\epsilon}\widehat{w_2}]_{j'}$, or deforms $[_iw_1\sigma_2^{\epsilon}\sigma_1^{\epsilon} w_2]_{j}$ into $[_iw_1\sigma_1^{-\epsilon}\widehat{w_2}]_{j'}$ for $\epsilon=\pm1$ and $\{j,j'\}=\{1,2\}$ when $w_2$ is a $3$-braid word, where $\widehat{w_2}$ is the flipped word of $w_2$.

Theorems & Definitions (56)

• Conjecture 1: Knot Entropy (KE) Conjecture Orl96Orl98
• Theorem 1
• Proposition 1: minsteptube
• Conjecture 2: KE Conjecture for Lattice Tubes BES19
• Proposition 2: BEISS18
• Proposition 3
• Corollary 1
• Theorem 2
• Theorem 3
• Proposition 4