Non-orientable slice surfaces and inscribed rectangles

We give a unified proof of existence of inscribed squares and inscribed rectangles with aspect ratio $\sqrt3$ in Jordan curves with a regularity condition, generalizing a result by Hugelmeyer. We also discuss differences between genera of smooth and locally-flat non-orientable surfaces in the 4-ball with boundary a given torus knot or 2-bridge knot. In particular, we establish that a result by Batson on the smooth non-orientable 4-genus of torus knots does not hold in the locally-flat category.