Problem 2 — Given a tuple ×N i=1(mi,ti) ∈ ZN, give a tractable expression in terms of Dehn- Thurston or other coordinates for the…

Definitions and mathematical statement

Fix the surface $F$ and the decomposition, reference arcs, and twist conventions used for Dehn--Thurston coordinates in the source. For an admissible integral coordinate tuple \[ u=((m_1,t_1),\ldots,(m_N,t_N)), \] let $\alpha(u)$ be the corresponding weighted family of pairwise disjoint curves and arcs. In the usual interior-pants-curve convention, $m_i$ is an intersection measure and $t_i$ a twist coordinate; boundary/arcs and zero-intersection cases require the source's additional conventions and admissibility conditions. Not every arbitrary integer tuple represents such a family.

Problem 2. Give a tractable, more nearly closed-form expression in these or other coordinates for the number of components of $\alpha(u)$. The source explicitly distinguishes this request from the known serial train-track-splitting algorithm that recovers a multiply weighted curve from an integral train-track measure.

The component-count convention must be fixed. If \[ \alpha(u)=w_1\alpha_1+\cdots+w_q\alpha_q \] is written with distinct nonisotopic support curves or arcs and positive integral weights, counting support components gives $q$, while interpreting weights as parallel multiplicities gives $\sum_{j=1}^q w_j$. These are different coordinate functions; the excerpt does not state which is meant or how parallel arcs are grouped. The target is an expression for the intended function $C(u)$ after that convention is recovered, not an invented replacement formula.

The source additionally asks for a description in Dehn--Thurston coordinates on $\operatorname{MF}(F)$ of the loci denoted $\mathcal A'(F)$ or $\operatorname{Arc}'(F)$. Here $\operatorname{MF}(F)$ is the relevant measured-foliation space, but the precise foliation convention and both primed loci are not defined in this record. It suggests quantum path-ordering techniques [5,6] or fermionic statistical physics [57,29] as possible approaches and refers to Problem 3. These are speculative methods and an unreproduced cross-reference, not a solution or a supplied computational complexity guarantee.

Short English statement

Find a tractable coordinate formula for the component count of an integral weighted curve-and-arc family, beyond reconstructing it by repeated train-track splitting. Clarify admissible coordinates and whether multiplicities or only distinct support components are counted.

Sources

• Local UnsolvedMath registry, permanent ID 11000132, source code catalogue entry; no proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, chapter context "Probing mapping class groups using arcs," Problem 2, PDF page 113; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29; the inventory distinguishes 329 labeled headings by source page.
• Source references [6] for splitting and [5,6,57,29] for suggested quantum/statistical techniques, without full bibliography entries. The research summary cites Penner--Harer, Combinatorics of Train Tracks (1992), as related background rather than as a supplied closed-form answer.