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We give a description of the moment graph for Bott-Samelson varieties inarbitrary Lie type. We use this, along with curve neighborhoods and explicitmoduli space computations, to compute a presentation for the small quantumcohomology ring of a particular Bott-Samelson variety in Type $A$. We also showthe conjecture $\mathcal{O}$ of Galkin, Golyshev, and Iritani holds for thatBott-Samelson variety.

The moment graph for Bott-Samelson varieties and applications to quantum cohomology