Seifert cohomology of trees

TL;DR

This paper introduces a cohomology theory for trees with connections to algebraic geometry and knot invariants, providing combinatorial descriptions and links to Heegaard-Floer homology.

Contribution

It defines a filtered cochain complex for trees, relates its cohomology to Alexander polynomials, and connects to Heegaard-Floer homology in the case of Dynkin diagrams.

Findings

Cohomology and spectral sequence have clear combinatorial descriptions.

Graded Euler characteristic matches Alexander polynomial for Dynkin diagrams.

Links established between tree cohomology and Heegaard-Floer homology.

Abstract

To every tree we associate a filtered cochain complex. Its cohomology and the corresponding spectral sequence have clear combinatorial description. If a tree is the Dynkin diagram of a simple plane curve singularity, the graded Euler characteristic of this complex coincides with the Alexander polynomial of the link. In this case we also point the relation to the Heegard-Floer homology theory, constructed by P. Ozsvath and Z. Szabo.