Minimal crossing diagrams of spatial graphs

TL;DR

This paper extends Tait's conjecture on minimal crossing diagrams to spatial graphs by developing adequacy-based frameworks for $1$-vertex graphs and for planar frameworks, using $all$-$A$ and $all$-$A^{-1}$ resolutions and tangle diagrams to connect graph diagrams to adequate link diagrams. It proves that adequate $1$-vertex graphs have minimal crossing number and that replacing edges and vertices of a planar framework by minimal-crossing components preserves minimality, thereby generalizing Sawollek's results to broader graph classes. The work also analyzes rigid vertex spatial graphs, providing a concrete example that answers a question of Adams et al. about minimal crossing diagrams under rigid isotopy. Collectively, the results give practical criteria for minimality across diverse spatial-graph families and illuminate how adequacy, planarity, and rigidity influence crossing number behavior.

Abstract

We prove that all $1$-vertex spatial graphs with adequate diagrams have minimal crossing number, and that spatial graph diagrams obtained by replacing vertices and edges of a planar embedded graph by minimal crossing link or spatial graph diagrams have minimal crossing number. Finally, we give an example in answer to a question of Adams et al. about minimal crossing diagrams of rigid vertex graphs.

Figures (13)

• Figure 1: A rational tangle can be written as $Q+H$ where $Q$ is trivial or is alternating and in the form on the left and $H$ is a (possibly trivial) row of horizontal twists.
• Figure 2: $A$- and $A^{-1}$-resolutions of a crossing and their associated grey segments.
• Figure 3: The smoothing in the center image is not reduced. On the right is a diagram with fewer crossings than the original.
• Figure 4: $G$ is isotopic to $G'$, but neither smoothing of $G$ is isotopic to a smoothing of $G'$.
• Figure 5: An R4 move taking $G$ to $G'$ induces an isotopy taking $L_T$ to $L_{T'}$.

Theorems & Definitions (23)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.1
• Definition 2.2
• Lemma 2.1
• proof
• Definition 2.3