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Word-length curve counting on the once-punctured torus

David Fisac, Mingkun Liu

TL;DR

This paper provides a complete combinatorial classification and exact counts for curves on the once-punctured torus with prescribed word length and self-intersection, focusing on zero and one self-intersection as well as arbitrary curves. It develops a necklace-based framework, including small-variation and 2-variation concepts, to translate curve counting into tractable combinatorial problems, aided by the Cohen–Lustig algorithm and Möbius inversion. The authors derive explicit formulas and asymptotics for primitive and nonprimitive curves, establishing parity-based counts for i=1 and linking simple-curve counts to necklace counts, with generating functions yielding closed forms for all reduced words. The results connect topological curve counting with explicit arithmetic and combinatorial structures (Euler totients, Möbius function, necklaces), enabling exact enumeration and asymptotic growth rates that illuminate the distribution of simple and self-intersecting curves on Σ_{1,1}.

Abstract

We classify closed curves on a once-punctured torus with a single self-intersection from a combinatorial perspective. We determine the number of closed curves with given word-length and with zero, one, and arbitrary self-intersections.

Word-length curve counting on the once-punctured torus

TL;DR

This paper provides a complete combinatorial classification and exact counts for curves on the once-punctured torus with prescribed word length and self-intersection, focusing on zero and one self-intersection as well as arbitrary curves. It develops a necklace-based framework, including small-variation and 2-variation concepts, to translate curve counting into tractable combinatorial problems, aided by the Cohen–Lustig algorithm and Möbius inversion. The authors derive explicit formulas and asymptotics for primitive and nonprimitive curves, establishing parity-based counts for i=1 and linking simple-curve counts to necklace counts, with generating functions yielding closed forms for all reduced words. The results connect topological curve counting with explicit arithmetic and combinatorial structures (Euler totients, Möbius function, necklaces), enabling exact enumeration and asymptotic growth rates that illuminate the distribution of simple and self-intersecting curves on Σ_{1,1}.

Abstract

We classify closed curves on a once-punctured torus with a single self-intersection from a combinatorial perspective. We determine the number of closed curves with given word-length and with zero, one, and arbitrary self-intersections.
Paper Structure (11 sections, 23 theorems, 67 equations, 7 figures)

This paper contains 11 sections, 23 theorems, 67 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Related work
  4. Simple curves
  5. Curves as necklaces
  6. Necklaces with small variation
  7. Counting
  8. Self-intersection one
  9. Characterization
  10. Counting
  11. All curves

Key Result

Theorem I.1

For any $L\in\mathbb{Z}_{\geq 4}$, we have where $\varphi$ stands for Euler's totient function.

Figures (7)

  • Figure 1: A once-punctured torus
  • Figure 2: Surgery on a word of the form $a b^{-1} a^i b a\omega$: remove the orange part and add the blue part.
  • Figure 3: Surgery on a word of the form $ab^{-1}a^{-i}b\omega$ with $i\geq2$
  • Figure 4: Possible words starting by $ab^{-1}a^{-1}b$
  • Figure 5: Words representing a curve with self-intersection one
  • ...and 2 more figures

Theorems & Definitions (47)

  • Theorem I.1
  • Corollary I.2
  • Corollary I.3
  • Theorem I.4
  • Theorem I.5
  • Corollary I.6
  • Corollary I.7
  • Theorem I.8
  • Corollary I.9
  • Remark I.10
  • ...and 37 more