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 triplygraded 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 triplygraded knot homology previously constructed by the authors. Thus we show that the corresponding superpolynomial 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 $HXY(L)$.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.06511
 Bibcode:
 2019arXiv190506511O
 Keywords:

 Mathematics  General Topology;
 Mathematics  Geometric Topology;
 Mathematics  Representation Theory
 EPrint:
 30 pages, no figures