Hardness of estimating normalized Betti numbers for Khovanov homology

Derive direct computational hardness results (e.g., DQC1-hardness or stronger) for approximating normalized Betti numbers β_{i,j}(K)/dim C_{i,j} of Khovanov homology, where the normalization factor is the chain space dimension dim C_{i,j} = \sum_{r : |r|=i} {\ell(r) \choose \frac{j-i+\ell(r)}{2}}.

Background

For simplicial complexes, approximating normalized Betti numbers is known to be DQC1-hard. The authors show several hardness results for additive approximations of Khovanov Betti numbers, but the normalized setting—natural for the original quantum homology algorithm—has not been established for Khovanov complexes. Proving such hardness would align Khovanov homology with known complexity barriers in qTDA.