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 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 inser..