On the Polytope Model and Near End node Isomorphisms of Type $A$ Kirillov--Reshetikhin Crystals

We prove an inductive formula to construct a path from the highest weight element to any given vertex in the crystal graph of the polytope realization of the Kirillov-Reshetikhin crystal $K{R}^{i,m}$ of type $A$. For $i\le 2$ or $i\ge n-1$, we provide explicit formulas of the same by only using the lowering crystal operators and in those cases, using these paths, we determine the explicit image of any element under the affine crystal isomorphisms between the polytope and the tableau realizations of the Kirillov-Reshetikhin crystals.