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For transverse oriented curves on an oriented surface, the algebraic intersection number is the sum of the local signs of their crossings. It depends only on their oriented homology classes and is antisymmetric:Its absolute value is at most the geometric intersection number.

Whenever $\alpha$ and $\beta$ bound a bigon, isotope one side across the bigon. The two removed crossings have opposite local signs: the induced directions around the two corners of an oriented disc are opposite. Thus the move reduces the geometric intersection number by two while leaving the algebraic intersection number of curves on an oriented surface unchanged.

Repeatedly remove bigons. The process terminates because the intersection count is a nonnegative integer, and the bigon criterion says that the resulting curves ${\alpha }_0$ and ${\beta }_0$ are in minimal position. Since every removal cancelled one positive and one negative crossing,

Choose an essential returning arc $\alpha$ based at $p$ and an essential simple closed curve $\gamma$ with $i(\alpha ,\gamma )>0$. The iteratesare again simple proper arcs based at $p$. Their geometric intersection number with a fixed transverse arc grows linearly with $|n|$, so they represent infinitely many isotopy classes. This is the standard Dehn twist construction.

There are exactly two $\mathrm{Mod}(S_{0,4})$-orbits of vertices. Equality or inequality of the two endpoints is preserved by every homeomorphism. Conversely, a homeomorphism can send any ordered configuration of punctures and complementary discs of an arc to any other of the same endpoint type. Thus all arcs joining distinct punctures lie in one orbit, and all arcs returning to one puncture lie in the other. These are the arc-complex vertex orbits of the four-punctured sphere.