Making Multicurves Cross Minimally on Surfaces

TL;DR

This work provides quasi-linear time algorithms for computing the geometric intersection number $i_S(\Gamma)$ and for producing a minimal-position representative $\Gamma'$ of a collection of closed curves on an orientable surface $S$, under input given as closed walks in a cellular embedding $M$. Building on reducing triangulations and the perturbation framework of Fulek–Tóth, the authors reduce the problem to hyperbolic and boundaryless settings, prove a new property ensuring $i_S(C)=i_{\Sigma}(C)$ for reduced walks on reducing triangulations, and then lift the results back to the original surface. The framework unifies surfaces without boundary, with boundary, and the torus, achieving $O(m + g^2 + g n \log(gn))$ time for $i_S(C)$ and producing perturbations of length $O(g^2 m n)$ in comparable time, while supporting multiple curves in minimal position. These results significantly improve prior quadratic and quartic bounds and expand applicability to multi-curve configurations with simpler proofs and broader generality. The approach leverages discrete geodesic-like properties from reducing triangulations and modern perturbation techniques to advance computational topology on surfaces.

Abstract

On an orientable surface $S$, consider a collection $Γ$ of closed curves. The (geometric) intersection number $i_S(Γ)$ is the minimum number of self-intersections that a collection $Γ'$ can have, where $Γ'$ results from a continuous deformation (homotopy) of $Γ$. We provide algorithms that compute $i_S(Γ)$ and such a $Γ'$, assuming that $Γ$ is given by a collection of closed walks of length $n$ in a graph $M$ cellularly embedded on $S$, in $O(n \log n)$ time when $M$ and $S$ are fixed. The state of the art is a paper of Despré and Lazarus [SoCG 2017, J. ACM 2019], who compute $i_S(Γ)$ in $O(n^2)$ time, and $Γ'$ in $O(n^4)$ time if $Γ$ is a single closed curve. Our result is more general since we can put an arbitrary number of closed curves in minimal position. Also, our algorithms are quasi-linear in $n$ instead of quadratic and quartic, and our proofs are simpler and shorter. We use techniques from two-dimensional topology and from the theory of hyperbolic surfaces. Most notably, we prove a new property of the reducing triangulations introduced by Colin de Verdière, Despré, and Dubois [SODA 2024], reducing our problem to the case of surfaces with boundary. As a key subroutine, we rely on an algorithm of Fulek and Tóth [JCO 2020].

Figures (5)

• Figure 2.1: (Top Left) An embedded graph $M$. (Top Right) The patch system $\Sigma$ of $M$. (Bottom Left) We usually think of $\Sigma$ as a "closed neighborhood" of $M$. (Bottom Right) The arcs of $\Sigma$.
• Figure 2.2: Removing a bigon in the proof of Lemma \ref{['lem:perturbations']}.
• Figure 3.1: (Left) A primitive closed curve $\hat{c}$, in minimal position. (Right) The fourth power $c$ of $\hat{c}$, in minimal position in a neighborhood of $\hat{c}$.
• Figure 4.1: (Left) The canonical system of loops $L$ for the surface $S$ of genus $g=4$. (Middle Left) A blue graph $Q$ on $S$, in general position with respect to $L$, that intersects every edge of $L$ at most $m \geq 0$ times. (Middle Right) A reducing triangulation $T$ whose 1-skeleton contains $L$. (Right) An embedding of $T$ on $S$ such that each edge of $T$ crosses $Q$ at most $O(gm)$ times. This construction trivially generalizes to higher genus.
• Figure 4.2: (Left) In the proof of Theorem \ref{['thm:main-without-boundary']}, a portion of $T$ is here represented in blue in some face of $Q$. The two black disk vertices belong to $M$. The two circle vertices were inserted in faces of $M$ to build $Q$, they are dual vertices. The dashed edge was deleted from $M$ to build $Q$. The plain edges belong to $Q$. (From Left to Middle) In the overlay $Q'$ between $T$ and $Q$, every edge of $Q$ is detached from its incident dual vertex, then contracted. (From Middle to Right) Some edge incident to the blue disk vertex of $T$ is contracted.

Theorems & Definitions (43)

• Theorem 1.1: Despré, Lazarus, 2019
• Theorem 1.2: Despré, Lazarus, 2019
• Theorem 1.3: Fulek, Tóth, 2020
• Theorem 1.4
• Proposition 1.4
• Lemma 2.1: untangling-graphs, Proposition 3.1 and Proposition 3.4
• Lemma 2.2: untangling-graphs, Proposition 3.7
• Lemma 2.3
• proof
• Lemma 2.4