Topology-Based Lattice Engineering in Condensed Soft Matter

In this study, we investigate the topological adaptability and structural resilience of periodic soft matter entanglements using the DNA tensegrity triangle, a foundational motif in structural DNA nanotechnology, as a model system. By simulating the Reidemeister moves from knot theory, which describe a series of “moves” by which the knot equivalence is preserved, we demonstrate that many variants of the tensegrity triangle maintain their lattice geometry, underscoring the motif’s inherent topological robustness. Using granular deformations in a series of closely related motifs, we systematically twist the helices and slide their ends relative to junction crossings to yield 48 distinct crystal structures. Notably, we Identify a novel poke-DX feature (PDX), which introduces rigid crossover configurations with enhanced crystallographic resolution and site-specific metal ion coordination. Further exploration reveals the formation of semi-junctions – a new class of four-arm junctions held together by a single rotatable bond, which feature relaxed torsional strain and altered crossover geometries. These configurations support lattice transformations into tetragonal and distorted rhombohedral forms as well as facilitate topological inversion between left- and right- handed triangles. Altogether, these findings illustrate how controlled topological operations at the molecular level can tune local flexibility and stiffness at key sites to affect long-range lattice geometry. This work positions DNA-based frameworks as a programmable platform for the design of architected materials, topological metamaterials, and nanoscale devices with tunable structural and functional properties.