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

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. • Linked regions in polymers occur in a localized way, like pearls on a string. • A first proof of knot and link localization for polymers in nanochannels. • Counts for fixed link type polygons grow asymptotically like decorated unknots. • Proofs for 4-plat unknotting via braid word insertions. • A first proof of a pattern theorem for unknot self-avoiding polygons.