  We modify our previous construction of link homology in order to include a
natural duality functor $\mathfrak{F}$.
  To a link $L$ we associate a triply-graded module $HXY(L)$ over the graded
polynomial ring $R(L)=\mathbb{C}[x_1,y_1,\dots,x_\ell,y_\ell]$. The module has
an involution $\mathfrak{F}$ that intertwines the Fourier transform on $R(L)$,
$\mathfrak{F}(x_i)=y_i$, $\mathfrak{F}(y_i)=x_i$.
  In the case when $\ell=1$ the module is free over $R(L)$ and specialization
to $x=y=0$ matches with the triply-graded knot homology previously constructed
by the authors. Thus we show that the corresponding super-polynomial satisfies
the categorical version of $q\to 1/q$ symmetry.
  We also construct an isotopy invariant of the closure of a dichromatic braid
and relate this invariant to $HXY(L)$.
