Topology plays a fundamental role in controlling the dynamics of adaptive biological and physical systems, from chromosomal DNA and biofilms to cilia carpets and worm collectives. Despite their long history, the subtle interplay between topology, geometry and mechanics in tangled elastic filaments remains poorly understood. To uncover the topological principles underlying the dynamics of knotted and tangled matter, we first develop a mapping between human-designed elastic knots and long-range ferromagnetic spin systems. This mapping gives rise to topological counting rules that predict the relative mechanical stability of commonly used climbing and sailing knots. Building upon this framework, we then examine the adaptive topological dynamics exhibited by California blackworms, which form living tangled structures in minutes but can rapidly untangle in milliseconds. Using blackworm locomotion datasets, we construct stochastic trajectory equations that explain how the dynamics of individual active filaments control their emergent topological state. By identifying the principles behind stability and adaptivity in living tangled matter, our results have applications in understanding broad classes of adaptive, self-optimizing biological systems.