Generating Series for Immersed Plane Curves

Definitions and mathematical statement

Let $g\in\mathbb Z_{\geq 0}$ denote the number of transverse self-intersections of an immersed curve in $\mathbb R^2$. The supplied statement does not specify the domain of the immersion or the precise equivalence relation meant by isotopy, so it does not yet determine a unique counting function.

For a fixed choice of admissible immersed curves and of isotopy equivalence, write \[ \mathcal C_g=\{\text{admissible immersed curves in }\mathbb R^2\text{ with only transverse self-intersections, exactly }g\text{ in number}\}/\!\sim, \qquad N(g)=\#\mathcal C_g. \] If every $N(g)$ is finite, the associated ordinary generating series is the formal power series \[ G(z)=\sum_{g=0}^{\infty}N(g)z^g\in\mathbb Z_{\geq 0}[[z]], \] where $z$ is a formal indeterminate; no analytic convergence is stipulated.

Problem, conditional on resolving the counting conventions. Find a combinatorial or geometric interpretation of $G(z)$ for the intended isotopy classes of immersed plane curves.

Unresolved qualifications. The record does not specify whether curves are closed, connected, parametrized, oriented, or rooted; whether intervals, noncompact domains, or several components are allowed; or what boundary or behavior-at-infinity conditions apply when relevant. It also does not specify whether isotopy means ambient isotopy of the plane acting on curve images or an equivalence of parametrized immersions, and whether orientation-reversing identifications are permitted. For a differentiable immersion $\gamma:M\to\mathbb R^2$, an ordinary transverse double point is an image point with exactly two distinct preimages $s,t\in M$, whose tangent images span $\mathbb R^2$. The record does not explicitly state whether all self-intersections must be ordinary double points, or how any higher multiple point would be counted. These choices must be settled before $\mathcal C_g$ and its finiteness are determined. The requested kind of interpretation is also not further specified.

Short English statement

Count isotopy classes of immersed plane curves by their number of transverse self-intersections, and seek a combinatorial or geometric meaning for the resulting ordinary generating series. The source leaves essential counting conventions unresolved, so this is a conditional statement requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30002513$, title Generating Series for Immersed Plane Curves, source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2014 and names no individual proposer.
• Supplied primary provenance: New Trends in Teichmüller Theory and Mapping Class Groups (2014), Oberwolfach Reports, pp. 395--452, DOI 10.4171/owr/2014/07. https://doi.org/10.4171/owr/2014/07. Also listed in the supplied literature sources as Oberwolfach Report 07/2014.
• Supplied literature source: Schaeffer--Zinn-Justin, plane curves. https://arxiv.org/abs/math-ph/0304034.
• Supplied literature source: Cahn--Fanoni--Petri, self-intersections. https://arxiv.org/abs/1603.00846.
• Supplied literature assessment, checked on exactly 2026-08-21: the record labels the problem open and reports partial enumeration results and conjectural asymptotics for immersed plane curves with prescribed transverse self-intersections, while describing the requested classification and generating-series interpretation as remaining open. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.