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2004 On the Asymptotic Number of Plane Curves and Alternating Knots
Gilles Schaeffer, Paul Zinn-Justin
Experiment. Math. 13(4): 483-493 (2004).


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 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 data strongly support our identification with a conformal field theory recently described by Read and Saleur.


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Gilles Schaeffer. Paul Zinn-Justin. "On the Asymptotic Number of Plane Curves and Alternating Knots." Experiment. Math. 13 (4) 483 - 493, 2004.


Published: 2004
First available in Project Euclid: 22 February 2005

zbMATH: 1063.05015
MathSciNet: MR2118273

Primary: 0516

Keywords: Asymptotic enumeration , planar maps , plane curves

Rights: Copyright © 2004 A K Peters, Ltd.


Vol.13 • No. 4 • 2004
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