Abstract

Let ${\mathrm{\Sigma }}_n$ denote the symmetric group on $n$ symbols and let ${\sigma }_n$ denote the standard $n$-cycle $(1,2,3,\dots <!--FUNCTION\; APPLICATION-->,n)$. We consider the combinatorial problem of counting $n$-cycles $\rho$ so that the commutator $[\rho ,{\sigma }_n]$ is again an $n$-cycle. With a constructive recipe, we generate factorially many such $n$-cycles for odd $n$ (when $n$ is even, it is easy to see that no such $n$-cycles exist). We apply this to counting certain square-tiled surfaces where $2g-1=n$ is the number of squares. Letting $S_g$ denote the closed, orientable surface of genus $g$, in joint work with Huang, Aougab constructed exponentially many (in $g$) mapping class group orbits of pairs of simple closed curves whose complement is a single topological disk. Our construction produces factorially many (again in $g$) such orbits. These new orbits additionally have the property that the absolute value of the algebraic intersection number is equal to the geometric intersection number, implying that each pair naturally gives rise to an origami.

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