Topology and geometry play fundamental roles in controlling the dynamics of biological and physical systems, from chromosomal DNA and biofilms to cilia carpets and worm collectives. How topological rules give rise to adaptive, self-optimizing dynamics in soft and living matter remains poorly understood. Here we investigate the interplay between topology, geometry and mechanics in knotted and tangled matter. We first examine the adaptive topological dynamics exhibited by California blackworms, which form disordered living tangles in minutes but can rapidly untangle in milliseconds. By combining link-based tangling metrics with stochastic trajectory equations, we explain how the dynamics of individual active filaments controls their emergent topological state. Building on this framework, we then investigate tangled structures with local alignment. We demonstrate how the algebra of braids governs the mechanics and stability of braided filamentous networks in a range of biological systems. By identifying how topology and adaptivity produce stable yet responsive structures, these results have applications in understanding broad classes of adaptive living systems.