Locked crossings are an innovation I introduced in basket making that allows making baskets from a spool of edge-notched ribbon, where the spool does not need to be passed through the work, and the shape of the basket is programmed by the positions of the notches. Locked crossings open a new possibility: programming the angular excess/deficit in each weave opening, thereby texturing an otherwise smooth basket into a miniature landscape of hills, dales, and saddles. I refer to this as corrugating the basket in the sense that improved stiffness is gained through patterned texturing. By reference to some results in knot theory, I explore the mathematical constraints on corrugated baskets that are made from a single, unicursal ribbon that closes into a loop. Models of the smallest such baskets, and a vase-scale maquette are illustrated.