Filaments and fibers represent a fundamental unit of biological matter, from chromosomal DNA and microtubule networks to cilia carpets and worm collectives. How topology and elasticity mediate coordination in biological filaments remains poorly understood. To uncover the topological principles at play in living matter, we first examine the adaptive topological dynamics exhibited by California blackworms, which form tangled aggregates in minutes but can rapidly untangle in milliseconds. By constructing stochastic trajectory equations, we build a predictive model which explains how the dynamics of individual active filaments controls their emergent topological state. We then examine the elastodynamics and coordination of individual filaments more closely, and exhibit mathematical principles underlying knot-formation in filamentous organisms across length scales. By identifying how topology and elasticity produce stable yet responsive structures, these results have applications in understanding organization and self-optimization in broad classes of biological systems.