2022/04/13 by Nicholas R. Beaton, Kai Ishihara, Beaton, Nicholas R. +11 · 1 voice
Computer Science · Engineering · Physics and Astronomy · #Composite Material Mechanics #Model Reduction and Neural Networks #Topological and Geometric Data Analysis
paper · doi:10.1016/j.dam.2025.08.042
openalex publication_date 2025/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/22
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 T ∗ , an ∞ × 2 × 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 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.