Dualizable link homology

Abstract

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 $HXY(L)$.

30 pages, no figures; comments are welcome

References in corpus (1)

• A categorification of a cyclotomic Hecke algebra

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• Algebra and geometry of link homology