Minimal geodesics of a torus with a hole

Let $R$ be a Riemann surface of genus $1$ with one hole. For a given homology class $\alpha\in H_1(R)$, the author determines that homotopy class within $\alpha$ which contains the shortest curve in $\alpha$. It turns out that this homotopy class is uniquely determined independently of the Riemannian metric. A conjecture of H. Cohn is thereby confirmed.
Bibliography: 7 titles.