Let $X=\mathbb C\setminus\{0,1\}$, equipped with the hyperbolic structure it inherits from Klein's modular $\lambda$ function $\lambda:\mathbb H \to X$. In each (non-peripheral and nontrivial) free-homotopy class of loop $S^1\to X$, there is a unique hyperbolic representative. For example, the figure-eight curve $\alpha$ with winding number $+1$ around $1$ and $-1$ around $0$, pictured below, has a unique hyperbolic geodesic representative.

In fact, the curve drawn in the above linked picture is precisely the path traced by the hyperbolic geodesic representative of $\alpha$, which is given by the lemniscate with equation $$16(x^2+y^2)((x-1)^2+y^2)=1.$$

*Proof sketch:* The conformal transformation $w(z)=(2z-1)^2$ is an orbifold covering map from $X$ to a hyperbolic orbifold $Y$, where $Y$ is $\mathbb C \setminus\{1\}$ with a single orbifold point at $0$ with angle $\pi$. The geodesic representative of $\alpha$ projects to a geodesic $\eta$ that begins on the orbifold point, travels once counterclockwise around $1$ and hits the orbifold point, then travels once clockwise around $1$ before finishing again at the orbifold point. The anticonformal automorphism $w\mapsto \frac {\bar{w}}{\bar{w}-1}$ is an orientation-reversing isometry of $Y$ that preserves the homotopy class of $\eta$, so it's not hard to see that $\eta$ must have image given by the fixed-point set of this isometry, namely $|w-1|^2=1$. Putting the pieces together, we find that the points in the image of the geodesic representative of $\alpha$ satisfy $|(2z-1)^2-1|^2=1$, which simplifies to the equation above.

**Question**: Are all closed hyperbolic geodesics on $X$ algebraic curves?

I admit that it is a bit rash to ask the question in this way — one could more modestly ask **which** hyperbolic geodesics have polynomial defining equations — but I do want to emphasize that I am not aware of any negative results about the defining equations (the other two figure-eight curves are easy to work out as above). Of course, it's possible that the lemniscate equation arises as some low-dimensional accident due to the special symmetry present in the figure-eight curve, but it also seems entirely plausible that all hyperbolic geodesics on $X$ are somehow very *convenient* from the viewpoint of the canonical complex coordinate of $X$.

I've tried just looking at other examples using Mathematica to see if I might guess some equation for them (Mathematica's built-in ModularLambda function makes it easy to draw hyperbolic geodesics on $X$), but perhaps I am not fluent enough in solution sets of rational equations in the plane to make anything work.

Two other comments:

I've tagged this question with "modular forms", because one could think of the above polynomial equation as a polynomial identity relating the real and imaginary parts of the modular function $\lambda$ along the hyperbolic geodesics in $\mathbb H$ that project to closed curves.

There is an obvious analogy with this very recent MathOverflow question, which is a knot theory version of the same idea.