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We obtain a presentation of principal subspaces of basic modules for thetwisted affine Kac-Moody Lie algebras of type $A_{2n-1}^{(2)}$, $D_n^{(2)}$ and$E_6^{(2)}$. Using this presentation, we construct exact sequences among theseprincipal subspaces, and use these exact sequences to obtain recursionssatisfied by graded dimensions of the principal subspaces. Solving theserecursions, we obtain the graded dimensions of the principal subspaces.

Vertex-algebraic structure of principal subspaces of the basic modules  for twisted Affine Kac-Moody Lie algebras of type $A_{2n-1}^{(2)}, D_n^{(2)},  E_6^{(2)}$