Further Study of Kanenobu Knots

We determine the rational Khovanov bigraded homology groups of all Kanenobu knots. Also, we determine the crossing number for all Kanenobu knots $K(p,q)$ with $pq>0$ or $\mathrm{\mid }pq\mathrm{\mid }\le \max \{\mathrm{\mid }p\mathrm{\mid },\mathrm{\mid }q\mathrm{\mid }\}$. In the case where $pq<0$ and $\mathrm{\mid }pq\mathrm{\mid }>\max \{\mathrm{\mid }p\mathrm{\mid },\mathrm{\mid }q\mathrm{\mid }\}$, we conjecture that the crossing number is $|p| + |q| + 8$.