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Khovanov homology \(\mathcal{H}(L)\) of a link \(L\) is a sequence of
\(\mathbb Z\)-graded abelian groups, whose Euler characteristic is
the Jones polynomial. In fact these groups are naturally graded by
\(\frac{1}{2}\mathbb{Z} \times \frac{1}{2}\mathbb{Z}\), but this does
not lead to new invariants: \(\mathcal{H}^{i,p,q}(L)\) is nontrivial
only for \(p=q\), half of the original \(\mathbb{Z}\)-grading. However,
this refined grading adds more structure to the homology, especially
its odd variant, which can be used to derive formulas for
the homology of composite links, such as split links or connected
sums, and to describe its module structure over the homology of
the unknot. These results are well-known for the classical Khovanov
homology, but they are new for its odd variant.
If time permits, I will apply the results to construct homological
operations, whose powers all are non-trivial.
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