Hypergraph Rotation Systems: A Unified Framework for Linked Knot Structures

We present a unified framework for the design of linked knot (LK) structures including links, knots, weaves, knits, and chainmail structures, extending them to articulated and compliant (spring-like) assemblies. The core of our approach is a combinatorial representation based on Hypergraph Rotation Systems (H-GRS) , in which topology is defined by the cyclic ordering of incidences between faces and edges. This formulation removes the need for manifold assumptions or global orientation, naturally supporting non-manifold connectivity. Building on this representation, we introduce integer-valued twist assignments on edges as local operators that deterministically control global cycle structure, linking, and connectivity. These operators enable the systematic construction of both unicursal (single-cycle) and multi-component configurations, which can exhibit different forms of compliant and articulated behavior. Unlike prior approaches based on manifold constraints or binary twisting, our framework supports multi-valued twisting and higher-order connectivity, enabling selective linking, partial merging, and articulated couplings not achievable in classical textile constructions. We demonstrate the framework across a wide range of structures and validate it through physical prototypes fabricated via 3D printing and paper-strip construction on a scaffold. By unifying combinatorial topology with the exploration of mechanical behavior, H-GRS provides a minimal and extensible foundation for the topology-driven design of linked filament structures, with applications in computational fabrication, architectured materials, and bio-inspired systems.