2018/10/19 by Shrestha, Sunrose T.
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1810.08687
Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of n squares and two cone points with angle 4 π each, we set up and parametrize the classification into four diagrams. Our main result is to provide formulae for enumeration of square-tiled surfaces in these four diagrams, completing the detailed count for genus two. The formulae are in terms of various well-studied arithmetic functions, enabling us to give asymptotics for each diagram using a new calculation for additive convolutions of divisor functions that was recently derived by the author and collaborators. Interestingly, two of the four cylinder diagrams occur with asymptotic density 1/4, but the other diagrams occur with different (and irrational) densities.