Consider the curves ₁, 2, : [0, 1] C, given by ₁(t) = ,

₂(t) = , and = ₁ - ₂.

Let z₀ C\([0, 1]). What values does the orbital ind(, z₀) take for different z₀?

First answer the question in words (using the intuitive geometric understanding of the orbital number). Then prove your classification. In your proof you may (as an extra challenge) write homotopies concretely in formulas, but you don't have to.

Hint: Start by sketching and consider for a few randomly chosen points z₀ on your sketch, what the orbital number is.

(Vt2 - 4,t) - (-2-4,t) (Vt2 - 4,t) - (-2-4,t)