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 therefore
$$
\langle\alpha_0,\beta_0\rangle
=\langle\alpha_1,\beta_0\rangle.
$$
This is the <isotopy invariance of algebraic intersection number>.

Solution

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

During a generic isotopy of one transverse curve, crossings with a fixed curve can only be created or removed in pairs of opposite local sign. Their signed sum, and hence the algebraic intersection number, is therefore invariant.


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