Bridge indices of spatial graphs and diagram colorings

TL;DR

This paper generalizes the Wirtinger number invariant from links to spatial graphs, proving its equality with the bridge index, and develops algorithms to compute these indices, leading to new insights into the structure of unknotted graphs.

Contribution

It extends the Wirtinger number to spatial graphs, proves its equivalence to the bridge index, and provides algorithms for estimating and computing these indices.

Findings

Wirtinger number equals the bridge index for spatial graphs.

Algorithm implementation improves estimation of bridge indices.

Existence of almost unknotted graphs with arbitrarily large bridge index.

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.