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Curves on the torus with few intersections

Igor Balla, Marek Filakovský, Bartłomiej Kielak, Daniel Kráľ, Niklas Schlomberg

TL;DR

Problem: determine the maximum size $N({\\mathbb T}^2,k)$ of a $k$-system of pairwise non-homotopic simple closed curves on the torus. Approach: translate the torus problem into counting $k$-nice subsets of $\\mathbb{Z}^2$ up to unimodular equivalence, prove geometric (area) and arithmetic (coprimality) bounds, bound the height via a linear program (LP$_{\\ell}$) and duals, and use computer-assisted enumeration for small $k$ to obtain a complete classification. Contributions: exact formulas for all $k$ (modulo a finite exceptional set) show $N({\\mathbb T}^2,k)=k+4$ for large $k$ with a precise mod-$6$ pattern, and globally $N({\\mathbb T}^2,k)\\le k+6$ with equality only for $k\\in\\{24,48,120,168\\}$; the work also resolves the column-number problem for generic two-row $\\Delta$-modular matrices via a two-row correspondence. Significance: bridges topology, discrete geometry, and integer programming by solving the torus $k$-system problem and its two-row $\\Delta$-modular column-number analogue.

Abstract

Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most k times, has a maximum size of $k+O(\sqrt{k}\log k)$. We determine the maximum size of such a set for every k. In particular, the maximum never exceeds k+6, and it does not exceed k+4 when k is large. As this quantity coincides with the maximal number of columns of a generic k-modular matrix with two rows, our result also settles the column number problem, a problem of interest in combinatorial optimization, for such matrices.

Curves on the torus with few intersections

TL;DR

Problem: determine the maximum size of a -system of pairwise non-homotopic simple closed curves on the torus. Approach: translate the torus problem into counting -nice subsets of up to unimodular equivalence, prove geometric (area) and arithmetic (coprimality) bounds, bound the height via a linear program (LP) and duals, and use computer-assisted enumeration for small to obtain a complete classification. Contributions: exact formulas for all (modulo a finite exceptional set) show for large with a precise mod- pattern, and globally with equality only for ; the work also resolves the column-number problem for generic two-row -modular matrices via a two-row correspondence. Significance: bridges topology, discrete geometry, and integer programming by solving the torus -system problem and its two-row -modular column-number analogue.

Abstract

Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most k times, has a maximum size of . We determine the maximum size of such a set for every k. In particular, the maximum never exceeds k+6, and it does not exceed k+4 when k is large. As this quantity coincides with the maximal number of columns of a generic k-modular matrix with two rows, our result also settles the column number problem, a problem of interest in combinatorial optimization, for such matrices.

Paper Structure

This paper contains 12 sections, 20 theorems, 79 equations, 4 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Column number problem
  3. Overview of the proof
  4. Preliminaries
  5. Nice sets with large height
  6. Bounding the height of a nice set
  7. Analysis of the linear program
  8. Upper bound on the height
  9. Nice sets with height at most three
  10. Algorithm
  11. Main result
  12. Conclusion

Key Result

Theorem 1

Let $K_0$ be the set containing the $59$ integers listed in Table tab:main. For every $k\in{\mathbb N}\setminus K _0$, it holds that The values of $N({\mathbb T}^2,k)$ for $k\in K_0$ are given in Table tab:main.

Figures (4)

  • Figure 1: Visualization of the relation of the matrices $A_5$ and $A_6$ in the proof of Lemma \ref{['lm:LP']}.
  • Figure 2: Visualization of a $24$-nice set with 30 elements presented in Section \ref{['sec:concl']}.
  • Figure 3: Visualization of a $48$-nice set with 54 elements presented in Section \ref{['sec:concl']}.
  • Figure 4: Visualization of a $120$-nice set with 126 elements presented in Section \ref{['sec:concl']}.

Theorems & Definitions (45)

  • Theorem 1
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Proposition 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 35 more