On the Asymptotic Number of Plane Curves and Alternating Knots

TL;DR

This paper proposes conjectures for the asymptotic growth of plane curves and prime alternating knots based on quantum field theory, supported by numerical tests using advanced sampling methods and linking to conformal field theory.

Contribution

It introduces novel conjectures connecting plane curve enumeration with quantum gravity and conformal field theory, supported by a new numerical testing approach.

Findings

Numerical data supports the conjecture linking plane curves to conformal field theory.

A new numerical test based on perfect samplers for large planar maps was developed.

The conjectures relate the asymptotic number of plane curves and prime alternating knots to universality classes in physics.

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 a random planar lattices, that are in turn interpreted as a model of 2d quantum gravity with matter. The identification of the universality class of this model yields the conjecture. Since approximate counting or sampling planar curves with more than a few dozens of intersections is an open problem, direct confrontation with numerical data yields no convincing indication on the correctness of our conjectures. However, our physical approach yields a more general conjecture about connected systems of curves. We take advantage of this to design an original and feasible numerical test, based on recent perfect samplers for large planar maps. The numerical datas strongly support our identification with a conformal field theory recently described by Read and Saleur.