We prove that, if a closed geodesic Γ on a complete finite type hyperbolic surface has at least 2 self-intersections, then the length of Γ has an lower bound 2log(5+2√6), and the lower bound is sharp, attained on a corkscrew geodesic on a thrice punctured sphere.
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Wujie Shen (2024) studied this question.
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