Topology is the mathematical study of the spatial properties that are preserved through the deformation, twisting and stretching of objects. Topological architectures are common in nature and can be seen, for example, in DNA molecules that condense and relax during cellular events(1). Synthetic topological nanostructures, such as catenanes and rotaxanes, have been engineered using supramolecular chemistry, but the fabrication of complex and reconfigurable structures remains challenging(2). Here, we show that DNA origami(3) can be used to assemble a Mobius strip, a topological ribbon-like structure that has only one side(4-6). In addition, we show that the DNA Mobius strip can be reconfigured through strand displacement(7) to create topological objects such as supercoiled ring and catenane structures. This DNA fold-and-cut strategy, analogous to Japanese kirigami(8), may be used to create and reconfigure programmable topological structures that are unprecedented in molecular engineering.