Abstract: We introduce the double Bott-Samelson cell associated to an arbitrary Kac-Moody group G and a pair of positive braids (b, d). We prove that the double Bott-Samelson cells are affine varieties whose coordinate rings are upper cluster algebras. We describe the Donaldson-Thomas transformation on double Bott-Samelson cells. As an application, we obtain a new geometric proof of Zamolodchikov’s periodicity conjecture in the cases of type $\Delta\otimes A$. If time permits, I will further talk about its connections with Knot theory and quantum groups. This is joint work with Daping Weng.