Twisted Edges: A Unified Framework for Designing Linked Knot (LK) Structures Using Labeled Non-Manifold Surface Meshes

We present Twisted Edges, a unified framework for designing Linked Knot (LK) structures using labeled non-manifold surface meshes. While the concept of edge twists, originating in topological graph theory, is foundational to these designs, prior approaches have been strictly limited to binary states. We identify this restriction as a critical barrier; binary twisting fails to capture the full spectrum of topological possibilities, rendering a vast class of structural and dynamic behaviors inaccessible. To overcome this limitation, we generalize the twist formulation to support arbitrary integer twist labels. This expansion reveals that while zero twists may introduce disconnections, applying even twists to 2-manifold meshes robustly preserves connectivity, transforming surfaces into fully connected, chainmail-like structures where faces form consistently linked cycles. Furthermore, we extend this framework to non-manifold meshes, where specific integer assignments prevent cycle merging. This capability, unattainable with binary methods, enables the design of partial connectivity and functional hinges, supporting dynamic folding and articulation. Theoretically, we show that these integer-twisted meshes correspond to knotted surfaces in four dimensions, with LK structures arising as their immersions into $\mathbb{R}^3$. By breaking the binary constraint, this work establishes a coherent paradigm for the systematic exploration of previously unstudied woven and articulated structures.