Late in the 19th century, A. A. Markoff initiated an extensive theory of the minima of indefinite binary quadratic forms, or, what is the same, extending Hurwitz's Theorem of diophantine approximation. He showed in particular that these minima begin with a countable discrete spectrum which monotonically increases to 3. Early the 20th century, work of L. E. Ford then implies that these values are related tothe geometry of the modular surface.
Some forty years later, H. Cohn recognized a connection between these initial values of Markoff's spectrum and certain closed geodesics on the so-called homology cover of the modular surface. In particular, the Markoff numbers, which comprise this initial countable set of values of the spectrum, correspond one-to-one to the simple closed geodesics on a hyperbolic once-punctured torus $\Gamma'\backslash mathcal{H}$ which is a six-fold cover of the modular surface. The same result holds if $\Gamma'$ is replaced by $\Gamma(3)$ or $\Gamma^3$. Next, C. Series suggested thenvestigation the once self-intersecting closed geodesics of these surfaces and their values in the Markoff spectrum.
This is exactly what D. Crisp and W. Moran did. They showed that there are two classes of these geodesics, and that the more interesting of these, their proper single self-intersecting or PSSI geodesics, have Markoff values quite low in the spectrum. These values are given by formulae virtually identical to those of the initial portion of the spectrum.
Crisp and Moran conjectured that these values were (also) isolated, and they established this for first 26 examples. This conjecture also has a charming geometry, especially on $\Gamma^3 \backslash mathcal{H}$, which permits one to come close to proving it. However here is , at the moment, an obstruction.