Abstract
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 4π 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. Interestingly, two of the four cylinder diagrams occur with asymptotic density 1/4, but the other diagrams occur with different (and irrational) densities.
Highlights
The main result of this paper is enumeration of the number of primitive square-tiled translation surfaces in the stratum H (1, 1) by their cylinder diagrams
Recall that a square-tiled translation surface is a closed orientable surface built out of unit-area axis-parallel Euclidean squares glued along edges via translations
A square-tiled surface is a closed orientable surface obtained from the union of finitely many Euclidean axis parallel unit area squares {∆1, . . . , ∆n} such that
Summary
The main result of this paper is enumeration of the number of primitive (connected) square-tiled translation surfaces in the stratum H (1, 1) by their cylinder diagrams. Let A(n), B(n), C(n) and D(n) count the number of primitive n-square surfaces in H (1, 1) with cylinder diagram A, B, C and D, respectively. The enumeration by cylinder diagram of primitive n-square surfaces in H (2) was done in unpublished work of Zmiaikou [19]. Let H(n) be the total number of primitive square-tiled surfaces in H (2). There are two cylinder diagrams for surfaces in H (2), which we can denote by F and G, and the number of primitive n-square surfaces with. Zmiaikou’s work, completing the enumeration of primitive square-tiled surfaces of genus 2 by cylinder diagram
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