Almost simple geodesics on the triply-punctured sphere

In this paper we study closed hyperbolic geodesics γ on the triply-punctured sphere M= C^ - { 0 , 1 , ∞} that are almost simple, in the sense that the difference δ= I(γ) - L(γ) between the self-intersection number of γ and its combinatorial (word) length is fixed. We show that for each fixed δ, the number of almost simple geodesics with L(γ) = L is given by a quadratic polynomial p _δ (L) , provided L≥ δ+ 4.