# Betti Numbers of the HOMFLYPT Homology

## TL;DR

This paper investigates the algebraic structure of the middle HOMFLYPT homology of links, focusing on Betti numbers, and demonstrates their role in understanding link properties and providing new obstructions to split links.

## Contribution

It introduces the study of Betti numbers of the middle HOMFLYPT homology, generalizing reduced HOMFLYPT homology and revealing new link invariants.

## Key findings

- Betti numbers are supported on a finite subset of Z^4 for each link.
- Betti numbers determine the Poincaré polynomial of the homology.
- Projective dimension is additive under split union of links.

## Abstract

In arXiv:math/0508510, 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 Poincar\'e 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.