Non-orientable slice surfaces and inscribed rectangles

Abstract

We discuss differences between genera of smooth and locally-flatnon-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 smoothnon-orientable 4-genus of torus knots does not hold in the locally-flat category.We further show that certain families of torus knots are not boundaries of embed-ded Möbius bands in the 4-ball and other 4 manifolds.Our investigation of non-orientable surfaces whose boundary is a giventorus knot is motivated by our approach to unify the proof of the existence of in-scribed squares and of inscribed rectangles with aspect ratio √3 in Jordan curveswith a regularity condition. This generalizes a result by Hugelmeyer for smoothJordan curves.