On the asymptotic number of plane curves and alternating knots

Abstract

We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of plane curves, a similar conjecture is made for the asymptotic number of prime alternating knots. The rationale leading to these conjectures is given by quantum field theory. Plane curves are viewed as configurations of loops on random planar lattices, that are in turn interpreted as a model of two-dimensional quantum gravity with matter. The identification of the universality class of this model yields the conjecture. Since approximate counting or samplin..