Bridge indices of spatial graphs and diagram colorings
arXiv:2410.23253 · doi:10.1016/j.disc.2025.114813
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.
arXiv comment38 pages, 36 figures, comments very welcome! v2: Final version, accepted for publication in Discrete Mathematics
References in corpus (9)
- On the distortion of knots on embedded surfaces
- Quandle coloring and cocycle invariants of composite knots and abelian extensions
- Reflection subgroups of Coxeter groups
- Wirtinger systems of generators of knot groups
- Stick index of knots and links in the cubic lattice
- Coxeter groups and meridional rank of links
- Additive invariants for knots, links and graphs in 3-manifolds
- Distortion and the bridge distance of knots
- New examples of Brunnian theta graphs