**Theorem 6.**

$$\mathcal{H}^{i}(\Delta^{2k+1}\mathbf{x}\_{1}) = \begin{cases} \mathbb{Z} \oplus \mathbb{Z} & i = 0 \\ 0 & i = 1 \\ 0 & i = 6k + 1 \end{cases}$$

,

**Proof.** The proof is similar to the proof of Theorem 5: Obtain all states, organized them in columns, assign a graded vector space to each state, form chain groups as a direct sum of all vector spaces along a column, and form the chain complex. Then, write the differential maps in terms of matrices using the ordered bases of the chain groups, and compute their kernels and images. Finally, find the Khovanov homology groups using the relation H*r*(*L*) = ker *dr* im*dr*+<sup>1</sup> .
