Spiraling and Folding: The Topological View

TL;DR

The paper constructs spiral-free and fold-free curve configurations with arbitrarily many intersections to challenge existing bounds on string graphs. It develops a rigorous torus-based framework using train tracks and Fibonacci-patterned intersections to produce pairs of curves that avoid $2$-spirals, then lifts these constructions to the plane via covering spaces to obtain planar spiral-free drawings with many intersections. A key contribution is a counterexample to Pach and Tóth, showing that a wide fold does not necessarily imply a spiral, thus exposing a gap in an claimed exponential bound on the complexity of minimal string-graph realizations. Collectively, the work demonstrates that removing spirals and folds is insufficient to bound complexity on torus or planar drawings, motivating the need for additional structural constraints in studying string graphs.

Abstract

For every $n$, we construct two curves in the plane that intersect at least $n$ times and do not form spirals. The construction is in three stages: we first exhibit closed curves on the torus that do not form double spirals, then arcs on the torus that do not form spirals, and finally pairs of planar arcs that do not form spirals. These curves provide a counterexample to a proof of Pach and Tóth concerning string graphs.

Figures (18)

• Figure 1: A bigon formed by $\alpha$ and $\beta$.
• Figure 2: Left: A spiral formed by $\alpha$ and $\beta$. Right: A fold formed by $\alpha$ and $\beta$. The vertical arc is a part of the curve $\alpha$, all the other arcs are parts of the curve $\beta$.
• Figure 3: A fold of width $3$ formed by $\alpha$ and $\beta$ in $A$.
• Figure 4: $\beta$ and $\gamma$ form a $1$-spiral. The dashed lines indicate that other subarcs may be present.
• Figure 5: $\beta$ passing through $R_1$ and $R_2$

Theorems & Definitions (31)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 4
• Remark 5
• Lemma 6
• proof