Bridge indices of spatial graphs and diagram colorings

TL;DR

This work extends the Wirtinger color-extension invariant from knots/links to spatial graphs and proves that the Wirtinger number ω(G) exactly equals the bridge index β(G). The authors provide a constructive Python algorithm to upper-bound β and combine it with algebraic lower bounds from quandle colorings and meridional-rank/Coxeter techniques to determine exact indices for large families of almost unknotted graphs. They also demonstrate that, for any negative Euler characteristic, there exist almost unknotted spatial graphs with arbitrarily large β, using clasping and vertex-sum constructions that preserve colorability. Collectively, the paper bridges combinatorial colorings and geometric complexity in spatial graphs and supplies practical tools for computing bridge indices in complex graph embeddings.

Abstract

We extend the Wirtinger number of links, an invariant originally defined by Blair, Kjuchukova, Velazquez, and Villanueva in terms of extending initial colorings of some strands of a diagram to the entire diagram, to spatial graphs. We prove that the Wirtinger number equals the bridge index of spatial graphs, and we implement an algorithm in Python which gives a more efficient way to estimate upper bounds of bridge indices. Combined with lower bounds from diagram colorings by elements from certain algebraic structures and clasping techniques, we obtain exact bridge indices for a large family of almost unknotted spatial graphs. We also show that for every possible negative Euler characteristic, there exist almost unknotted graphs of arbitrarily large bridge index.

Figures (36)

• Figure 1: A trivial tangle, which contains some number of trees and arcs (not necessarily grouped together as shown). We did not include vertices where the tangle intersects $\partial B$, since when we glue two trivial tangles together along these vertices, we think of the resulting closed spatial graph as no longer having vertices where we glued.
• Figure 2: When we recast a spatial graph diagram as a planar graph, we delete small neighborhoods of the undercrossings and add endpoints to each side of the deleted portion.
• Figure 3: Some examples (blue and green) and non-examples (red) of strands. The blue strand has weight $3$ and the green strand has weight $2$.
• Figure 4: The coloring move: if the overstrand is colored and one of the understrands at the crossing is colored (with possibly the same color), then the move extends the coloring to the other understrand.
• Figure 5: This diagram is $\tfrac{7}{2}$-Wirtinger colorable, using three seeds with weights $2$ (red), $3$ (blue), and $2$ (green).

Theorems & Definitions (62)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Example 2.1
• Example 2.2
• Definition 2.3
• Definition 3.1
• Proposition 3.1
• proof