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

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 ${\mathbb{T}}^{\ast }$, 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 ${\mathbb{T}}^{\ast }$ 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.

Rearranged, with some figures updated. A much shorter version without proofs has been published as a letter at https://doi.org/10.1088/1751-8121/ad6c01

References in corpus (2)

• Knot probabilities in equilateral random polygons
• A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting

Cited by in corpus (1)

• Random knotting in very long off-lattice self-avoiding polygons