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Let G 4 be the space of all simple closed geodesics on the punctured sphere Σ 4 . We construct an explicit homeomorphism of the completion of G 4 onto a circle by using geometric intersection numbers. Also, we relate these geometric intersection numbers to trace polynomials of transformations corresponding to geodesics in G 4 in a representation of π 1 (Σ 4 ) into P SL(2, C).doi:10.1090/s1088-4173-97-00020-9 fatcat:xg7hdospljewphz3vk63g7sdz4