Let T be the once-punctured torus, let C(T) denote the set of closed curves on T, and let O be any mapping class group orbit in C(T). For every word length n, determine exactly the number N_O(n)=#gamma∈O:ell(gamma)=n, where ell(gamma) is the word length used for curves on the once-punctured torus. Chas's conjecture asserts that these orbit-by-orbit counts admit the claimed exact formulas for every orbit and every word length.

Chas’s conjecture on exact curve counting

Let $T$ be the once-punctured torus, let $\mathcal{C}(T)$ denote the set of closed curves on $T$ , and let $\mathcal{O}$ be any mapping class group orbit in $\mathcal{C}(T)$ . For every word length $n$ , determine exactly the number $N_{\mathcal{O}}(n)=\#\{\gamma\in\mathcal{O}:\ell(\gamma)=n\}$ , where $\ell(\gamma)$ is the word length used for curves on the once-punctured torus. Chas's conjecture asserts that these orbit-by-orbit counts admit the claimed exact formulas for every orbit and every word length.

References

Primary source

arXiv

Additional references

Exact curve counting of given word length on the once-punctured torus — arXiv

geometric-topology mapping-class-groups

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed preprint claims to settle the conjecture by giving exact counts for every curve length, but the claim has not been independently verified.

Chas’s conjecture concerns fine, orbit-by-orbit enumeration of curves on a once-punctured torus. The latest preprint claims exact counting at every word length.

Known results

Fisac and Liu (2024) classified primitive curves with one self-intersection and obtained exact word-length counts, as well as formulas for all closed curves.
Earlier work established exact orbit counts for self-intersection numbers $0$ and $1$ , with partial results for $2$ .

September 2026 claimed solution

A September 2026 arXiv preprint, Exact curve counting of given word length on the once-punctured torus, claims exact orbit-by-orbit counting at every word length, which would settle Chas’s conjecture. The claim is unverified.

Current status (as of September 2026): A preprint claims a complete solution, but the conjecture remains unverified pending independent checking.

Sources

No solutions have been posted yet.