Some fast algorithms for curves in surfaces

TL;DR

This work develops polynomial-time algorithms for fundamental surface-topology tasks involving curves: computing the geometric intersection number $i(C_1,C_2)$ and testing isotopy for normal 1-manifolds, with bounds polynomial in the surface presentation size and logarithmic in curve weights. Central to the approach are normal/standard forms, handle-structure formalisms, and the Agol–Hass–Thurston algorithm to manage pairings and their orbits, complemented by fast normalisation, minimal-position, and cutting techniques that keep complexity under control. The authors extend these techniques to patterns on surfaces, defining generalized isotopy regions and Reeb-type moves to place curves in locally minimal positions relative to patterns, which has prospective applications in 3-manifold hierarchies. Overall, the paper provides a cohesive, algorithmic framework for efficient topological analysis of curves on surfaces with practical impact for computational topology and geometric group theory.

Abstract

We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ and $C_2$ in a compact orientable surface $S$. The surface $S$ is presented via a triangulation or a handle structure, and the 1-manifolds are given in normal form via their normal coordinates. The running time is bounded above by a polynomial function of the number of triangles in the triangulation (or the number of handles in the handle structure), and the logarithm of the weight of $C_1$ and $C_2$. This algorithm represents an improvement over previous work, since its running time depends polynomially on the size of the triangulation of $S$ and it can deal with closed surfaces, unlike many earlier algorithms. Another algorithm, with similar bounds on its running time, can determine whether $C_1$ and $C_2$ are isotopic. We also present a closely related algorithm that can be used to place a standard 1-manifold into normal form.

Figures (16)

• Figure 1: A bigon and a half-bigon for two properly embedded 1-manifolds $C_1$ and $C_2$
• Figure 2: A standard curve in a triangulated surface, with a simplifying disc $D$
• Figure 3: Shown is a system of arcs in triangle. This system is specified by picking orientations on the edges, as shown, and setting $n_1 = 9$, $n_2 = 4$ and $n_3 = 1$. The arcs are given by three pairings. For example, one set of arcs is given by the pairing $[2, 3] \cap \mathbb{Z} \rightarrow [4,5] \cap \mathbb{Z}$, sending $x$ to $7-x$, and where the domain and range are both viewed as subsets of $[1,n_1] \cap \mathbb{Z}$.
• Figure 4: The three types of a simplifying disc for a 1-manifold in a surface. On the left is an interior-simplifying disc. In the middle and on the right are boundary-simplifying discs
• Figure 5: Parallelity handles of $S \backslash\backslash C$

Theorems & Definitions (118)

• Theorem 1.1: Geometric intersection number
• Theorem 1.2: Isotopy of 1-manifolds
• Theorem 1.3: Normalise standard 1-manifold in a triangulation
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Definition 2.6