  We compute the factorisation homology of the four-punctured sphere and
punctured torus over the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$
explicitly as categories of equivariant modules using the framework of
`Integrating Quantum Groups over Surfaces' by Ben-Zvi, Brochier, and Jordan. We
identify the algebra of invariants (quantum global sections) with the spherical
double affine Hecke algebra of type $(C^\vee_1,C_1)$, in the four-punctured
sphere case, and with the `cyclic deformation' of $U(su_2)$ in the punctured
torus case. In both cases, we give an identification with the corresponding
quantum Teichm\"uller space as proposed by Teschner and Vartanov as a
quantization of the moduli space of flat connections.
