A sign assignment in totally twisted Khovanov homology

Abstract

We lift the characteristic-2 totally twisted Khovanov homology of Roberts and Jaeger to a theory with $\mathbb{Z}$ coefficients. The result is a complex computing reduced odd Khovanov homology for knots. This complex is equivalent to a spanning-tree complex whose differential is explicit modulo a sign ambiguity coming from the need to choose a sign assignment in the definition of odd Khovanov homology.