Unfocused notes on the Markoff equation and T-Singularities

We consider minimal resolutions of the singularities for weighted projective planes of type \(\mathbb{P}(e^2, f^2, g^2)\), where \(e, f, g\) satisfy the Markoff equation \( e^2 + f^2 + g^2 = 3efg\). We give a complete classification of such resolutions in terms of continued fractions similar to classical work of Frobenius. In particular, we investigate the behaviour of resolutions under mutations and describe a Cantor set emerging as limits of continued fractions.