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Exact curve counting of given word length on the once-punctured torus
Filippo Baroni, David Fisac, Mingkun Liu
TL;DR
The paper studies exact curve counting by word length on the once-punctured torus, where word length offers the prospect of a closed counting expression. It establishes a theorem with finitely many parameters and an explicit periodic function, supported by word-stability and train-track methods, while giving computable bounds for the initial threshold.
Problem
Word length offers the prospect of a closed expression for curve counting, unlike hyperbolic lengths.
Method
The paper uses word-stability transformations and pre-train-track combinatorics to organize curves and derive the counting formula.
Results
C(L) is an explicit periodic function with period at most ∏_{i=1}^N lcm(P_i, Q_i), and |C(L)| ≤ 2^N; m is 2 with non-trivial torsion and 4 otherwise.
Takeaways & Limitations
The counting can be expressed using finitely many coefficients found by the algorithm in Section 5.
Takeaways & Limitations
The initial threshold L0 receives a trivial bound, although it is computable and generically closer to the orbit's minimal word length.
Abstract
from arXiv · showhide
On the once-punctured torus, we give an exact formula for the number of curves in any given mapping class group orbit of given word length. This settles a conjecture of Chas in [Cha16].
1 Introduction
The paper gives an exact word-length counting formula for every mapping class group orbit of essential curves on the once-punctured torus, settling Chas’s conjecture. Its formula combines generalized Euler totient terms with bounded periodic corrections, derived from stable positive-suborbit trees and length-preserving resolutions.
- Main result: Theorem 1.1 gives an exact formula for the number of orbit curves of each sufficiently large word length, with coefficients, a threshold L0, and a periodic correction.The correction has period at most the least common multiple of the relevant parameters, and its magnitude is bounded by twice the number of terms.
- Counting formula: The counting formula uses functions φP,Q(L) that count coprime positive solutions to Px + Qy = L, generalizing Euler’s totient function.For fixed P and Q, these counts are primitive lattice points on the corresponding first-quadrant segment and vary quasilinearly.
- Examples and computation: The algorithm computes the counting formula in exponential time in the minimal orbit word length and verifies all examples previously computed by Chas.The authors report runtimes below one second for writable examples.
- Counting formula: Low-intersection counting functions are linear combinations of Euler totient functions, while higher-intersection examples require weighted generalizations or periodic corrections.Theorem 1.1 guarantees that every type can be expressed in the stated general form.
- Setting and organization: The mapping class group is isomorphic to SL2(Z), generated by Dehn twists L and R, whose positive actions organize the relevant suborbits as binary rooted trees.Every orbit element reaches one of the two positive suborbits after applying a power of the order-four element S.
- Proof strategy: The proof decomposes positive suborbits into stable regions where word length grows linearly, with finitely many unstable curves outside the stable subtrees.Each maximal stable subtree is forward-complete, and its interior admits a length-preserving resolution.
- Main result: The main result settles Chas’s conjecture by extending explicit counting functions beyond the specific low-intersection cases previously known.Earlier formulas had been proved or verified only for selected low-intersection types.
2 Word stability
The section characterizes stable curves through admissible word forms and train-track carrying, then uses decreasing complexity to decompose orbit trees and organize mapping-class representatives.
- Stable words: A stable curve has a representative built from nonzero commutator powers separated by nonempty monotone words.The commutator is z = [a, B], and monotonicity constrains the signs of generator exponents.
- Complexity reduction: For any essential curve, only finitely many elements of its positive suborbit are non-stable.The proof defines a complexity c([w]) and shows that applying L or R strictly decreases it after repushing non-admissible subwords.
- Train-tracks: Stability is equivalent to being carried by the pre-train-track τ, with no illegal turns.The track’s incoming and outgoing half-edges are cyclically consecutive, and carrying means a representative loop lies on τ without illegal turns.
- Admissibility: Every cyclically reduced cyclic word can be rewritten into admissible form through finitely many curve-preserving movements, without creating zZ or Zz.The rewriting process decreases the relevant defect measure η, which vanishes exactly for admissible words.
- Orbit organization: Every orbit element enters one of the two positive suborbits after applying a power of the order-four elliptic element S.The full orbit count is m times the count in their union, with m = 2 when the stabilizer has nontrivial torsion and m = 4 otherwise.
3 Counting
Counting proceeds by analyzing maximal stable subtrees, where cancellation patterns stabilize and length grows linearly. The resulting exact formulas combine coprime-solution counts with bounded periodic corrections.
- Stable subtrees: A maximal stable subtree is a rooted binary tree generated by L- and R-actions from a stable root.The counting function depends on the regularity of cancellation among monotone subwords and commutators.
- Cancellation patterns: Cancellation counts are constant on the interior and may differ only along the two boundary rays.These constants are denoted c_int(w), c_L(w), and c_R(w).
- Exact formulas: For sufficiently large L, each maximal stable subtree has a counting function of the form φP,Q(L − 4K) + C(L), with C(L) periodic.Here P, Q ≥ 1, K ≥ 0, and the period of C(L) is at most lcm(P, Q), with C(L) ∈ {−2, …, 2}.
- Exact formulas: The main term φP,Q counts coprime positive solutions of Px + Qy = L, while the correction balances boundary-ray cancellation differences.The solution count varies quasilinearly as L grows.
- Example: For γ = Ja2b2A2b2AbK and L ≥ 12, the total orbit count is an explicit sum of five generalized totient terms plus a period-12 correction.The correction values are [0, 0, 0, 1, 0, −2, 0, 2, 0, −1, 0, 0] modulo 12.
- Algorithm and scope: The initial threshold L0 is computable but has only a trivial exponential upper bound in the minimal orbit length.Each L- or R-action can at most double length, yielding the stated coarse bound.
4 Intersection resolutions
The section defines equivariant local intersection resolutions and connects them to word combinatorics. These resolutions identify length-preserving behavior inside maximal stable subtrees.
- Local surgeries: At each intersection point, two local smoothings produce multiloops with lower intersection number.The operations extend from representatives to free homotopy classes and mapping-class orbits.
- Equivariance: Resolution maps are equivariant under the Mod(T)-action, so the same surgery operation is defined consistently across an entire orbit.The proof interprets resolution as a Whitehead-type move on train-tracks.
- Word combinatorics: Linking-pair equivalence classes encode intersection points, including self-intersections, for curves represented by words.The correspondence uses cyclic shifts, inversions, and equivalence relations on letter positions.
- Length preservation: At least one resolution preserves word length because it reorders letters without introducing cancellations.This length-preserving property is established for every intersection in the resolution framework.
- Self-intersections: For a self-intersection splitting w = uv, one resolution gives {JuK, JvK} and the other gives Juv−1K.For nonprimitive curves, the corresponding operations split powers or cancel powers.
- Stable-subtree interpretation: Each maximal stable subtree is characterized as a maximal connected, forward-complete subtree where a full resolution is length-preserving in its interior.Finiteness of the resolution list ensures one resolution is length-preserving on infinitely many curves in the stable subtree.
5 Algorithm
The algorithm converts cyclically reduced words into terminal words through commutator rewrites, cancellations, and mapping-class actions, then computes the counting formula from the resulting list. Termination is guaranteed by a strictly decreasing complexity, while terminal words are stable enough for the formula to apply.
- Algorithm: A terminal word is a cyclic word avoiding the listed cancellation subwords, including aA, aB, Aa, Ab, bA, bB, Ba, and Bb.
- Algorithm: The procedure repeatedly applies full and partial commutator moves, cancellation moves, and L,R-actions, also restarting from S(w).Terminal outputs are deduplicated before the counting formula is computed.
- Correctness: Every terminal word contains no zZ or Zz subwords and is therefore a stable word, allowing the counting formula to be computed on each terminal subtree.
- Termination: Full and partial commutator moves strictly decrease the complexity µ(w) = µL(w) + µR(w).The decrease is established separately for full moves and for partial moves when no full move is available.
- Termination: The recursion depth is at most µ(w) ≤ ℓ(JwK), so the algorithm terminates after finitely many recursive stages.
- Correctness: Allowed words remain allowed under full and partial commutator moves, while branch words remain allowed after L∗ and R∗.