Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 3 / d / Solution

On a genus-two surface, let $\alpha$ be a separating simple closed curve that cuts the surface into two once-punctured tori. Choose a simple closed curve $\beta$ that passes from one side to the other and back, with the two crossings arranged in minimal position. The crossings have opposite signs, sobut the bigon criterion shows that they cannot be removed and henceEquivalently, $\alpha$ is separating and therefore represents zero in first homology, forcing its algebraic intersection with every curve to vanish even though its geometric intersection need not vanish.