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Cross-linked filament bundles, such as in cilia and flagella, are ubiquitousin biology. They are considered in textbooks as simple filaments with largerstiffness. Recent observations of flagellar counterbend, however, show thatinduction of curvature in one section of a passive flagellum instigates acompensatory counter-curvature elsewhere, exposing the intricate role of thediminutive cross-linking proteins at large-scales. We show that this effect, amaterial property of the cross-linking mechanics, modifies the bundle dynamicsnon-trivially, and induces a bimodal $L^2-L^3$ length-dependent materialresponse that departs from the Euler-Bernoulli theory. Hence, the use ofsimpler theories to analyse experiments can result in paradoxicalinterpretations. Remarkably, the counterbend dynamics instigates counter-wavesin opposition to driven oscillations in distant parts of the bundle, withpotential impact on the regulation of flagellar bending waves. These resultshave a range of physical and biological applications, including the empiricaldisentanglement of material quantities via counterbend dynamics.

The counterbend dynamics of cross-linked filament bundles and flagella