Narrow Results

We prove that the degree of the Hilbert polynomial of the HOMFLYPT homology of a closed braid $B$ is $l-1$, where $l$ is the number of components of $B$. This controls the growth of the HOMFLYPT homology with respect to its polynomial grading. The Hilbert polynomial also reveals a link polynomial hidden in the HOMFLYPT polynomial.

In [Rasmussen, Khovanov–Rozansky homology of two-bridge knots and links, Duke Math. J.136 (2007) 551–583], Rasmussen observed that the Khovanov–Rozansky homology of a link is a finitely generated module over the polynomial ring generated by the components of this link. In the current paper, we study the module structure of the middle HOMFLYPT homology, especially, the Betti numbers of this module. For each link, these Betti numbers are supported on a finite subset of ${\mathbb{Z}}^4$. One can easily recover from these Betti numbers the Poincaré polynomial of the middle HOMFLYPT homology. We explain why the Betti numbers can be viewed as a generalization of the reduced HOMFLYPT homology of knots. As an application, we prove that the projective dimension of the middle HOMFLYPT homology is additive under split union of links and provides a new obstruction to split links.

We apply the Rasmussen spectral sequence to prove that the ${\mathbb{Z}}^3$-graded vector space structure of the HOMFLYPT homology over ${\mathbb{Z}}_2$ detects unlinks. Our proof relies on a theorem of Batson and Seed stating that the ${\mathbb{Z}}^2$-graded vector space structure of the Khovanov homology over ${\mathbb{Z}}_2$ detects unlinks.