A spectrum connecting the bridge index and the braid index

Artin introduced the braid in 1925 to study links: he restricted a link to wrapping around an axis in only one direction, which constrained its behavior enough to facilitate the development of powerful tools. In 2024, Aranda, Binns and Doig extended the idea of the braid to the booklink by allowing the link to reverse the direction of its wrapping in a controlled way, which enables many tools from braid theory to be extended to generic links. In this work, we study an invariant developed from booklinks, the bridge–braid spectrum of a link, which interpolates between the classical bridge and braid indices. It illuminates the way booklinks organize link embeddings into a spectrum with pure braids at one extreme and plat diagrams at the other. We apply foliation theory to understand the structures of split and composite links and to generate decomposition formulas for their booklink spectra. We then generate a table for the spectra of prime knots up to nine crossings.