Fabrication of Topologically Complex Three-Dimensional Microstructures: Metallic Microknots

This paper describes a method for fabricating three-dimensional (3D) microstructures with complex topologiestrefoil, figure eight, and cinquefoil knots, a chain with complex links, Borromean rings, a Mobius strip, and a torus. This method is based on the strategy of decomposing these structures into figures that can be printed on the surfaces of cylinders and planes that contact one another. Any knot can be considered as a pattern of crossings of lines, in which one line crosses “over” or “under” the second. We map these “over” and “under” crossings onto the surface of a cylinder and show that only two cylinders, in tangential contact (with axes parallel) and with lines allowed to cross from the surface of one to the surface of the second, are required to make any knot (or, in a combinatorial mathematical sense, any graph) in a topography that consists only of smooth curves. To form free-standing metal microstructures, we begin by printing appropriate patterns onto a continuous metal film on two cylinders...