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A celebrated and deep theorem in the theory of Riemann surfaces states theexistence and uniqueness of the Jenkins-Strebel differentials on a Riemannsurface under some conditions, but the proof is non-constructive and examplesare difficult to find. This paper deals with an example of a simple case,namely Jenkins-Strebel differentials on the Riemann sphere with four fixedsimple poles. We will give explicit expressions of these Jenkins-Strebeldifferentials by means of the Weierstrass $\wp$ function and expose a simplealgorithm determining the correspondence between these differentials and someclasses of simple closed curves on the Riemann sphere with four points removed.

Jenkins-Strebel Differentials on the Riemann Sphere with Four Simple Poles