Checkerboard graph monodromies (Q2314868)

This is an important paper on the structure of positive braid knots. Thanking Peter Feller ``for confusing us with fake cycles'', the authors provide a common extension of the classes of prime positive braid knots and positive tree-like Hopf plumbings, proving also that the link type of a prime positive braid closure is determined by its associated linking graph. Readers are treated to a thrilling romp through brick diagrams, linking graphs, open books, checkerboard graphs and Coxeter words, with related Lemmas, Corollaries and Propositions cross-referenced as Theorems 3 through 10. Included is a comparison of oriented links arising from checkerboard trees with Bonahon-Siebenmann's unoriented arborescent links [\textit{F. Bonahon} and \textit{L. Siebenmann}, ``New Geometric Splittings of Classical Knots, and the Classification and Symmetries of Arborescent Knots'', Preprint, available at \url{https://dornsife.usc.edu/francis-bonahon/preprints/}] (in connection with which readers should note those authors' inaccurate account of Conway's 1970 tabulation of non-alternating 11 crossing knots), a reference to the authors' list of positive prime braid knots of genus 6 or less (said to be in complete agreement with one compiled by Stoimenow) and some challenging remaining problems involving checkerboard trees, graphs and graph links.