Dualizable link homology

TL;DR

This paper introduces a duality-enhanced triply-graded link homology that exhibits a q↔1/q symmetry and extends previous knot homology theories, providing new invariants for links and their braids.

Contribution

It develops a modified link homology with a natural duality functor, connecting it to existing theories and establishing new isotopy invariants for dichromatic braids.

Findings

The homology module has an involution intertwining Fourier transform.

Specialization recovers previous triply-graded knot homology.

Super-polynomial satisfies categorical q↔1/q symmetry.

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 $HXY(L)$.