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

TL;DR

This paper proves the Knot Entropy Conjecture for lattice polygons in a tube, establishing asymptotic growth rates for polygons with fixed knot types and providing new insights into polymer modeling and knot theory.

Contribution

It provides the first proof of the Knot Entropy Conjecture for knots and links in a lattice tube, linking asymptotics of polygons to unknot counts and establishing bounds using braid insertions and transfer-matrices.

Findings

Established asymptotic bounds for polygons with fixed link-type in a lattice tube.

Proved polygons can be unknotted via braid insertions.

Derived new combinatorics results for lattice polygons and knot theory.

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.