Hao Wu
Published 2017-03-21Version 1
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 polynomials of both the middle and the reduced HOMFLYPT homologies. 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.
Comments: 15 pages, 2 figures
Sutured manifolds and $L^2$-Betti numbers
Irrational pencils and Betti numbers
Varieties of general type with the same Betti numbers as $\mathbb P^1\times \mathbb P^1\times\ldots\times \mathbb P^1$
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