Odd Khovanov Homology for Hyperplane Arrangements

TL;DR

This paper introduces homology theories for hyperplane arrangements that categorify classical polynomial invariants and generalize odd Khovanov homology to signed arrangements, establishing invariance under hyperplane Reidemeister moves.

Contribution

It defines new homology theories for hyperplane arrangements, extending odd Khovanov homology and proving invariance under generalized Reidemeister moves.

Findings

Homology theories categorify characteristic, Poincare, and Tutte polynomials.

Chain complexes exhibit long-exact sequences and dg-algebra structures.

Invariance of chain homotopy type under hyperplane Reidemeister moves.

Abstract

We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletion-restriction triples and dg-algebra structures. We also consider signed hyperplane arrangements, and generalize the odd Khovanov homology of Ozsvath-Rasmussen-Szabo from link projections to signed arrangements. We define hyperplane Reidemeister moves which generalize the usual Reidemeister moves from framed link projections to signed arrangements, and prove that the chain homotopy type associated to a signed arrangement is invariant under hyperplane Reidemeister moves.