Counting curves on surfaces
Norman Do, Musashi A. Koyama, Daniel V. Mathews

TL;DR
This paper explores an elementary combinatorial problem of counting disjoint arcs on surfaces with boundary points, revealing a rich structure connected to topological recursion, algebraic geometry, and mathematical physics.
Contribution
It introduces a new combinatorial enumeration problem for curves on surfaces, demonstrating recursive formulas, quasi-polynomial behavior, and links to advanced geometric and physical theories.
Findings
Curve counts follow an effective recursion via topological recursion.
Enumerative counts exhibit quasi-polynomial behavior.
Generating functions show structures akin to free energies and partition functions.
Abstract
In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface , and fix a number of points on its boundary. We ask: how many configurations of disjoint arcs are there on whose boundary is ? We find that this enumerative problem, counting curves on surfaces, has a rich structure. For instance, we show that the curve counts obey an effective recursion, in the general framework of topological recursion. Moreover, they exhibit quasi-polynomial behaviour. This "elementary curve-counting" is in fact related to a more advanced notion of "curve-counting" from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative…
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Taxonomy
TopicsMathematical Dynamics and Fractals · Stochastic processes and statistical mechanics · Geometric and Algebraic Topology
