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

Choose a generic isotopy from ${\alpha }_0$ to ${\alpha }_1$ while keeping ${\beta }_0$ fixed. Except at finitely many times, all intersections are transverse. At each exceptional time a tangency creates or removes a pair of crossings. In oriented local coordinates the two crossings have opposite signs, so their contributions cancel. The signed sum is constant throughout the isotopy, and thereforeThis is the isotopy invariance of algebraic intersection number.