Toeplitz asked whether every Jordan curve in the Euclidean plane contains 4 points that form the corners of a square? More generally, what about the corners of a rectangle with prescribed aspect ratio? The latter question was recently answered by Greene-Lobb for smooth Jordan curves, but even the original question remains open in general. We follow a strategy of Hugelmeyer to relate Toeplitz's question to the following variant of a classical knot concordance question: For a given knot K in the 3-sphere S3 (or S^2xS1), what is the smallest integer among the first Betti numbers of non-orientable surfaces in the 4-ball B4 (or B^3xS1) with boundary K?

Based on joint work with Marco Golla.