Khovanov’s conjecture over Z[c

1.1. Khovanov’s Homology and Conjecture. In his 1999 paper [K1] in Duke Mathematical Journal, Mikhail Khovanov showed that the Jones link polynomial is the graded Euler characteristic of a bigraded homology module H (D) over Z[c], associated to a diagram D of the link. H (D) is the homology of a bigraded chain complex C(D) with a (1, 0)-bigraded differential. He also explained how each link cobordism induces a homomorphism between the homology modules of its boundary links, and conjectured that this homomorphism would be invariant up to sign under ambient isotopy of the link cobordism. In [J] we proved that, if formulated precisely, this conjecture is indeed true, under the condition that the indeterminate c = 0, i.e. when the modules are abelian groups. (This case suffices to retrieve the Jones polynomial as the Euler characteristic.) In [K2], Khovanov gave a different proof of this fact, and, in the introduction, renewed his conjecture in the general case.