Symmetries of equivariant Khovanov homology

We study symmetries in equivariant versions of Khovanov homology, which include (i) the construction of an involution $\widehat{\sigma }$ for the $U(2)$-equivariant theory, (ii) an integral lifting $\widehat{\nu }$ of the Shumakovitch operation $\nu$, and (iii) splitting of the $U(1)$- and $U(1)\times U(1)$-equivariant theories generalizing earlier work over ${\mathbb{F}}_2$. Finally, we relate these structures to the Rasmussen $s$-invariant over an arbitrary field $F$.