Localization behaviour in a phenomenological model of three-dimensional knots

Abstract

We propose to model three-dimensional (3D) knots as effective vertex networks with vanishing topological exponents to study their equilibrium behaviour. This model is self-consistent and predicts weak localization in knots with up to five essential crossings. The resulting localization exponent for the 3D trefoil is in numerical agreement with a recent simulation study. In more complex knots, however, delocalization is expected. It is shown that this approach corresponds to the decomposition of the knot into C coupled loops of variable size, where C is the number of essential crossings.