On the Minimum Area of Null Homotopies of Curves Traced Twice

We provide an efficient algorithm to compute the minimum area of a homotopy between two closed plane curves, given that they divide the plane into finite number of regions. For any positive real number $\varepsilon>0$, we construct a closed plane curve $\gamma$ such that the minimum area of a null homotopy of $2\cdot\gamma$ is less than $\varepsilon$ times that of $\gamma$. We also establish a lower bound on how complex a desired closed curve has to be.