Odd Khovanov homology

Abstract

We describe an invariant of links in $S^3$ which is closely related to Khovanov’s Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov’s definition with an exterior algebra. The two invariants have the same reduction modulo $2$, but differ over $\mathbb{Q}$. There is a reduced version which is a link invariant whose graded Euler characteristic is the normalized Jones polynomial.