The most symmetric once punctured hyperbolic tori are parametrized as T(t) by a real parameter t>2, which will be written as t=2+x. The trace of a closed geodesic on T(t) is given by a polynomial in x. In joint work with Xiangfei Li, we investigate the properties of these polynomials and the ordering of the lengths of closed geodesics on T(t). We make conjectures and prove some of them; in the case where x=1, we obtain some conjectured properties of the Markoff numbers.

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