Covering space maps for $n$-point functions with three long twists

We consider correlation functions in symmetric product orbifold CFTs on the sphere, focusing on the case where all operators are single-cycle twists, and the covering surface is also a sphere. We directly construct the general class of covering space maps where there are three twists of arbitrary lengths, along with any number of twist-2 insertions. These are written as a ratio of sums of Jacobi polynomials with $ΔN+1$ coefficients $b_N$. These coefficients have a scaling symmetry $b_N\rightarrow λb_N$, making them naturally valued in $\mathbb{CP}^{ΔN}$. We explore limits where various ramified points on the cover approach each other, which are understood as crossing channel specific OPE limits, and find that these limits are defined by algebraic varieties of $\mathbb{CP}^{ΔN}$. We compute the expressions needed to calculate the group element representative correlation functions for bare twists. Specializing to the cases $ΔN=1,2$, we find closed form for these expressions which define four- and five-point functions of bare twists.