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1. The problem statement, all variables and given/known data

Apparently if [itex]\gamma_4 = \gamma_2 +\gamma_3 -\gamma_1-\gamma_3[/itex], then [itex]\gamma_4[/itex] is homotopic to [itex]\gamma_5[/itex] in any region containing [itex]\gamma_1[/itex],[itex]\gamma_2[/itex], and the region between them minus z.

I am not convinced that this is true.

I can picture how you could transform [itex]\gamma_4[/itex] into [itex]\gamma_5[/itex] one another by first moving the two [itex]\gamma_3[/itex]s apart, then kind of going around the circle and finally contracting the curve, but I am not convinced that this is a continuous function from the unit interval cross the unit interval. Specifically, I am not convinced that you can just move the two [itex]\gamma_3[/itex]s apart in a continuous way. How can you rigorously show that there is a homotopy?

2. Relevant equations

3. The attempt at a solution

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# Homotopic curves

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