Escape from the second dimension: A topological distinction between edge and screw dislocations

Volterra's definition of dislocations in crystals geometrically distinguishes edge and screw defects according to whether the Burgers vector is perpendicular or parallel to the defect. A homotopy-theoretic analysis of dislocations as topological defects fails to differentiate edge and screw. Here we bridge the gap between the geometric and topological descriptions by demonstrating that there is a topological difference between screw and edge defects. Our construction distinguishes edge and screw based on the disclination-line pairs at the core of smectic dislocations. By exploiting the connection between topology and geometry in the form of Gaussian curvature, this analysis results in an invariant for dislocations in the saddle-splay vector. This construction can be generalized to crystals with triply periodic order.